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+{"text":"\\section{Introduction}\nLet $H$ be a real separable Hilbert space and $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ be a stochastic basis satisfiying the usual conditions, $L=(L(t))_{t\\geq 0}$ be a square-integrable cylindrical L\\'evy process in a real separable Hilbert space $U$ with respect to the stochastic basis $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$, taking values in a possibly larger Hilbert space $U_1\\supset U$, and $B:U\\to H$ is a bounded linear operator. Consider an $H$-valued stochastic convolution process\n\\begin{equation}\\label{eq:X}\nX(t)=E(t)X_0+\\int_0^tE(t-s)B\\,\\ensuremath{\\mathrm{d}} L(s)\n\\end{equation}\nwhere $(E(t))_{t\\in [0,T]}$ is a family of bounded linear operators on $H$ and $X_0$ is an $\\cF_0$-measurable $H$-valued random variable. Without loss of generality, all Hilbert spaces are assumed to be infinite-dimensional.\nImportant examples of such processes are weak solutions $(X(t))_{t\\geq0}$ of certain stochastic partial differential equations (SPDEs, for short) driven by additive L\\'{e}vy noise; these can be written as abstract It\\^o stochastic differential equations\n\\begin{equation}\\label{eq:mainEq}\n\\ensuremath{\\mathrm{d}} X(t)+AX(t)\\,\\ensuremath{\\mathrm{d}} t=B\\,\\ensuremath{\\mathrm{d}} L(t),\\quad t\\geq0;\\quad X(0)=X_0,\n\\end{equation}\nwhere $-A$ is the generator of a strongly continuous semigroup $(E(t))_{t\\geq 0}$ on $H$. In particular, we consider the stochastic heat equation\n\\begin{equation}\\label{eq:SHE}\n\\ensuremath{\\mathrm{d}} X(t)+\\Lambda X(t)\\ensuremath{\\mathrm{d}} t= \\ensuremath{\\mathrm{d}} L(t),\\quad t\\geq0;\\quad X(0)=X_0,\n\\end{equation}\nand the stochastic wave equation, written as a first order system,\n\\begin{equation}\\label{eq:SWE}\n\\begin{aligned}\n\\ensuremath{\\mathrm{d}} X_1(t) -X_2(t)\\ensuremath{\\mathrm{d}} t&=0, &&t\\geq0;\\quad X_1(0)=X_{0,1},\\\\\n\\ensuremath{\\mathrm{d}} X_2(t)+\\Lambda X_1(t)\\ensuremath{\\mathrm{d}} t&= \\ensuremath{\\mathrm{d}} L(t),&&t\\geq0;\\quad X_2(0)=X_{0,2}.\n\\end{aligned}\n\\end{equation}\nIn both cases $\\Lambda:=-\\Delta=-\\sum_{j=1}^d\\partial^2\/\\partial\\xi_j^2$ is the Laplace operator on $L^2(\\ensuremath{\\mathcal{O}})$ where $\\ensuremath{\\mathcal{O}}\\subset\\bR^d$ is a bounded domain.\nFor the precise abstract setup of these equations we refer to Sections \\ref{sec:Heat} and \\ref{sec:Wave}.\nIn general, however, we do not require that $(E(t))_{t\\geq 0}$ enjoys the semigroup property so that the abstract framework can accommodate Volterra-type evolution equations as well, see Subsection \\ref{subsec:sve} for details.\n\nConsider an\napproximation $\\tilde X=(\\tilde X(t))_{t\\in [0,T]}$ of the process $(X(t))_{t\\in[0,T]}$ given by\n\\begin{equation}\\label{eq:Xtilde}\n\\tilde X(t)=\\tilde E(t)X_0+\\int_0^t\\tilde E(t-s)B\\,\\ensuremath{\\mathrm{d}} L(s),\n\\end{equation}\nwhere $(\\tilde E(t))_{t\\in[0,T]}$ is a family of bounded linear operators on $H$, which is again not necessarily (extendable to) an operator semigroup. For example, the family  $(\\tilde E(t))_{t\\in[0,T]}$ may be a time-interpolated solution operator family of a space-time discretized stochastic evolution problem, when $H$ is an $L^2$-space of some spatial domain $\\ensuremath{\\mathcal{O}}$. We study the so-called weak error\n\\begin{equation}\\label{eq:defWeakError}\ne(T):=\\bE\\big(G(\\tilde X(T))-G(X(T))\\big)\n\\end{equation}\nfor suitable test functions $G:H\\to \\bR$. At the heart of the paper are the error representation formulae for $e(T)$, Theorem \\ref{thm:errorRep} and Corollary \\ref{cor:errorRep}. The proof of Theorem \\ref{thm:errorRep} is\nbased on Kolmogorov's backward equation for the martingale $Y(t)=E(T)X_0+\\int_0^tE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)$, $t\\in[0,T]$, which has the important property that $Y(T)=X(T)$. The introduction of such an auxiliary process $Y$ is well-known for equations with Gaussian noise and has been used by many authors in a weak error analysis, see, for example \\cite{debussche_printems,KovLarLin11,KovLarLin13,KovPri14b} to mention just a few (compare also \\cite{CJK14, DPJR12}). However, the extension of those arguments is not straightforward and the resulting error representation formula differs from the one in the Gaussian case in \\cite{KovLarLin13}. One of the difficulties in the general L\\'evy case (in contrast to the Gaussian case) is that there are no readily available, sufficiently general results on  Kolmogorov's backward equation to suit our analysis. We remedy this, at least for $Y$ as above, in Proposition \\ref{prop:backwardKolm}. Another complication arises from the fact that we use tools from the theory of stochastic integration based on two different settings. One, where we integrate operator-valued processes w.r.t.\\ a Hilbert space-valued L\\'evy process, promoted in the monographs \\cite[Chapter 8]{PesZab07}, \\cite{Met82,MetPel80}, and another one where we integrate Hilbert space-valued integrands w.r.t.\\ a Poisson random measure \\cite{MRT13,Pre10}. The problem occurs because our setting for stochastic differential equations is based on the first approach while the proof of the error representation formula is based on an It\\^o formula which appears in \\cite[Theorem 3.6]{MRT13}; the latter form is well suited for our purposes, but it is formulated using the second approach for stochastic integration. Therefore, in the appendix we link the two stochastic integrals so that we can use the results from both theories.\n\nUsing the abstract error representation we study the weak error of  space-time discretizations for parabolic equations, such as the stochastic heat equation and a stochastic Volterra integro-differential equation, and a hyperbolic equation, the stochastic wave equation. As space discretization we employ a standard continuous finite element method. For the stochastic heat equation we use the backward Euler method, for the Volterra integro-differential equation the backward Euler method combined with a convolution quadrature, and for the stochastic wave equation an $I$-stable rational approximation of the exponential function, such as the Crank-Nicolson scheme, as time integrators. For all equations considered here, the Hilbert-Schmidt norm condition $$\\nnrm{\\Lambda^{(\\beta-1\/\\rho)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(L^2(\\ensuremath{\\mathcal{O}}))}<\\infty, \\quad \\beta>0,$$ determines the rate of convergence, where $\\rho=1$ for the heat and wave equations and $\\rho\\in (1,2)$ for the Volterra equation. Here $U=L^2(\\ensuremath{\\mathcal{O}})$ and $Q\\in\\ensuremath{\\mathscr L}(L^2(\\ensuremath{\\mathcal{O}}))$ is the covariance operator of $L$ as introduced in Section~\\ref{sec:DrivingProcess}.\n\nFor the stochastic heat equation, we show in Theorem \\ref{thm:SHE} that for twice continuously differentiable test functions with bounded second derivatives the rate of weak convergence is essentially twice that of strong convergence.  It is at least $O(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta)$, $\\beta\\in(0,1]$, modulo a logarithmic term, where $h$ and $\\ensuremath{\\Delta t}$ are the space- and time-discretization parameters, respectively. This extends the corresponding result from \\cite{LinSch13}, where, in contrast to the present paper, the analysis is restricted to so-called impulsive cylindrical processes on $L^2(\\ensuremath{\\mathcal{O}})$ as driving noise. Moreover, there is a serious restriction on the jump size intensity measure in \\cite[Section~6]{LinSch13}\nadmitting only processes of bounded variation (on finite time intervals).\nHere, the only restriction we have on $L$ is that it is square-integrable, non-Gaussian and has mean zero. Furthermore, we also remove the boundedness assumption on the test functions and their first derivative.\n\nIn Subsection \\ref{subsec:sve} we briefly discuss a stochastic Volterra-type integro-differential equation and obtain a weak rate of order at least $O(h^{2\\beta}+(\\ensuremath{\\Delta t})^{\\rho\\beta})$, $\\beta\\in(0,1\/\\rho]$, where $\\rho$ depends on the order of the convolution kernel appearing in the equation.\n\n\nFor the stochastic wave equation the additional technical condition \\eqref{eq:assgSWE} has to be imposed in order to prove that the weak order is twice the strong order. The order of weak convergence is found to be at least $O(h^{\\min(r, 2\\beta\\frac{r}{r+1})}+(\\ensuremath{\\Delta t})^{\\min(1,2\\beta\\frac{p}{p+1})})$, see Theorem~\\ref{thm:SWE}. Here $p$ and $r$ are the classical orders of the time-discretization and of the finite element method. We would like to point out that, while the extra condition \\eqref{eq:assgSWE} on the second derivative on the test function is restrictive, it trivially holds for the important function $g(x)=\\|x\\|_{L^2(\\ensuremath{\\mathcal{O}})}^2$. We also give a sufficient condition \\eqref{eq:weqii} for the above rate of weak convergence to hold which only involves the jump intensity measure of $L$ and $\\Lambda$, with no further restriction on the test functions. Condition  \\eqref{eq:weqii} is sufficient for $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(L^2(\\ensuremath{\\mathcal{O}}))}<\\infty$ and, in special cases, it is even equivalent to it, see Example~\\ref{ex:spec}.\nFurthermore, as far as the authors know, there are no results available in the literature concerning weak approximation of hyperbolic stochastic partial differential equations driven by L\\'evy noise.\n\nLet us remark that weak error estimates for approximations of L\\'{e}vy-driven stochastic ordinary differential equations have been considered by various authors, see, e.g.~\\cite{JKMP05, MikZha11, PlaBru10, ProTal97} and the references therein. There also exists a series of papers on strong error estimates for approximations of SPDEs  driven by L\\'{e}vy processes or Poisson random measures, see, for example \\cite{Bar10, BarLan12a, BarLan12b, DunHauPro12, Hau08, HauMar06, Lan10} and compare also Remarks \\ref{rem:strongErrorSHE} and \\ref{rem:strongErrorSWE} below. However, to the best of our knowledge, the first steps in a \\emph{weak} error analysis for L\\'{e}vy-driven SPDEs have been done only recently in the already mentioned article \\cite{LinSch13}.\n\nThe present paper\nis organized as follows. In Section \\ref{sec:Setting and preliminaries} we describe the abstract framework of the paper, introduce infinite-dimensional L\\'evy processes with several examples and a framework for linear stochastic partial differential equations driven by additive L\\'evy noise. Assumption \\ref{ass:abstractSetting} summarizes the main assumptions for the general setting of the paper. In Section \\ref{sec:An error representation formula} we state and prove two representation formulae, Theorem~\\ref{thm:errorRep} and Corollary~\\ref{cor:errorRep},  for $e(T)$ given by \\eqref{eq:defWeakError}. The main ingredient in their proofs is Proposition~\\ref{prop:backwardKolm} on Kolmogorov's backward equation. We use the representation formula from Corollary \\ref{cor:errorRep} to establish weak convergence rates for a space-time discretization scheme first for parabolic equations in Section \\ref{sec:Heat}, where we consider the stochastic heat equation (Subsection~\\ref{subsec:she}) and, in less detail, a stochastic Volterra-type integro-differential equation (Subsection~\\ref{subsec:sve}). Then, in Section \\ref{sec:Wave}, we analyse the weak error of a numerical scheme for the stochastic wave equation. In the appendix we link stochastic integration with respect to Poisson random measures to integration with respect to infinite-dimensional L\\'evy processes.\n\n\n\n\n\n\n\n\n\\section{Setting and preliminaries}\n\\label{sec:Setting and preliminaries}\n\nHere we describe in detail our abstract setting and collect some background material from infinite-dimensional stochastic analysis.\\\\\n\n\\noindent\n{\\bf General notation.}\nLet $(\\cH,\\langle\\arg,\\arg\\rangle_{\\cH})$ and $(\\cG,\\langle\\arg,\\arg\\rangle_{\\cG})$ be real, separable Hilbert spaces and denote by $\\ensuremath{\\mathscr L}(\\cH,\\cG)$, $\\ensuremath{\\mathscr L_1}(\\cH,\\cG)$ and $\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ the spaces of linear and bounded operators, nuclear operators and Hilbert-Schmidt operators from $\\cH$ to $\\cG$, respectively.  The corresonding norms are denoted by $\\nnrm{\\arg}{\\ensuremath{\\mathscr L}(\\cH,\\cG)}$, $\\nnrm{\\arg}{\\ensuremath{\\mathscr L_1}(\\cH,\\cG)}$ and $\\nnrm{\\arg}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}$. If $\\cH=\\cG$, we write $\\ensuremath{\\mathscr L}(\\cH)$, $\\ensuremath{\\mathscr L_1}(\\cH)$ and $\\ensuremath{\\mathscr L_2}(\\cH)$ instead of $\\ensuremath{\\mathscr L}(\\cH,\\cH)$, $\\ensuremath{\\mathscr L_1}(\\cH,\\cH)$ and $\\ensuremath{\\mathscr L_2}(\\cH,\\cH)$. Given a measure space $(M,\\cM,\\mu)$ and $1\\leq p <\\infty$, we denote by $L^p(M;\\cH)=L^p(M,\\cM,\\mu;\\cH)$ the space of all $\\cM\/\\cB(\\cH)$-measurable mappings $f:M\\to\\cH$ with finite norm $\\nnrm{f}{L^p(M;\\cH)}=(\\bE\\nnrm{f}{\\cH}^p)^{1\/p}$, where $\\cB(\\cH)$ denotes the Borel $\\sigma$-algebra on the Hilbert space $\\cH$. By $C^n(\\cH,\\bR)$ we denote the space of all $n$-times continuously Fr\\'{e}chet differentiable functions $f:\\cH\\to\\bR,\\,x\\mapsto f(x)$. By $C_{\\text b}^n(\\cH,\\bR)$ we denote the subspace of functions from $C^n(\\cH,\\bR)$ which are bounded together with their derivatives. Identifying $\\cH$ and $\\ensuremath{\\mathscr L}(\\cH,\\bR)$ via the Riesz isomorphism, we consider for fixed $x\\in\\cH$ the first derivative  $f'(x)$ as an element of $\\cH$. Similarly, the second derivative $f''(x)$ is considered as an element of $\\ensuremath{\\mathscr L}(\\cH)$.\nWe also write $f_x$ and $f_{xx}$ instead of $f'$ and $f''$.\n\n\n\\subsection{The driving L\\'{e}vy process $L$}\n\\label{sec:DrivingProcess}\n\nThe process $L=(L(t))_{t\\geq0}$ in Eq.~\\eqref{eq:mainEq} is a  L\\'{e}vy process with values in a real and separable Hilbert space $U_1$, defined on a filtered probability space $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ satisfying the usual conditions (cf.~\\cite{PesZab07}).\n$L$ is $(\\cF_t)$-adapted and for $t,h\\geq0$ the increment $L(t+h)-L(t)$ is independent of $\\cF_t$. We always consider a c\\`{a}dl\\`{a}g\n(right continuous with left limits) modification of $L$, i.e., a modification such that $L(t)=\\lim_{s\\searrow t}L(s)$ for all $t\\geq0$ and $L(t-):=\\lim_{s\\nearrow t}L(s)$ exists for all $t>0$, where the limits are pathwise limits in $U_1$.\nOur standard reference for Hilbert space-valued L\\'{e}vy processes is \\cite{PesZab07}.\n\nIn order to keep the exposition simple, we assume that $L$ is square-integrable, i.e., $\\bE\\nnrm{L(t)}{U_1}^2<\\infty$, and that the Gaussian part of $L$ vanishes. Moreover, we assume that $L$ has mean zero, i.e., $\\bE L(t)=0$ in $U_1$.\nLet $\\nu$ be the jump intensity measure (L\\'{e}vy measure) of $L$. Note that the jump intensity measure $\\nu$ of a general L\\'{e}vy process in $U_1$ satisfies $\\nu(\\{0\\})=0$ and $\\int_{U_1}\\min(1,\\nnrm{y}{U_1}^2)\\nu(dy)<\\infty$, cf.~\\cite[Section 4]{PesZab07}. Due to our assumptions we have\n\\begin{equation}\\label{eq:assnu}\n\\int_{U_1}\\nnrm{y}{U_1}^2\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty,\n\\end{equation}\nand the characteristic function of $L$ is given by\n\\begin{equation}\\label{eq:charFunctionL}\n\\bE e^{i\\langle x,L(t)\\rangle_{U_1}}=\\exp\\Big\\{-t\\int_{U_1}\\big(1-e^{i\\langle x,y\\rangle_{U_1}}+i\\langle x,y\\rangle_{U_1}\\big)\\nu(\\ensuremath{\\mathrm{d}} y)\\Big\\},\\quad t\\geq0,\\;x\\in U_1.\n\\end{equation}\nConversely, any $U_1$-valued L\\'{e}vy process $L$ satisfying \\eqref{eq:assnu} and \\eqref{eq:charFunctionL} is square-integrable, with mean zero and vanishing Gaussian part.\n\n\n\nLet $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ be the covariance operator of $L$.\nIt is determined by the jump intensity measure $\\nu$ via\n\\begin{equation}\\label{eq:Q1nu}\n\\langle Q_1x,y\\rangle_{U_1}=\\int_{U_1}\\langle x,z\\rangle_{U_1}\\langle y,z\\rangle_{U_1}\\nu(\\ensuremath{\\mathrm{d}} z),\n\\quad x,y\\in U_1,\n\\end{equation}\nsee~\\cite[Theorem 4.47]{PesZab07}.\nFurther, let\n\\begin{equation*}\n(U_0,\\langle\\arg,\\arg\\rangle_{U_0}):=\\big(Q_1^{1\/2}(U_1),\\langle Q_1^{-1\/2}\\arg,Q_1^{-1\/2}\\arg\\rangle_{U_1}\\big)\n\\end{equation*}\nbe the reproducing kernel Hilbert space of $L$, where $Q_1^{-1\/2}$ denotes the pseudo-inverse of $Q_1^{1\/2}$, see~\\cite[Section 7]{PesZab07}. Recall that the operator $B$ in Eq.~\\eqref{eq:mainEq} is defined on the Hilbert space $U$. We assume that\n\\begin{equation}\\label{eq:U0subsetU}\nU_0\\subset U\\subset U_1,\n\\end{equation}\nand that the inclusions \\eqref{eq:U0subsetU} define continuous embeddings. We denote the embedding of $U_0$ into $U$ by $J_0\\in \\ensuremath{\\mathscr L}(U_0,U)$ and set\n\\begin{equation}\\label{eq:defQ}\nQ:=J_0J_0^*\\in\\ensuremath{\\mathscr L}(U).\n\\end{equation}\nThe nonnegative and symmetric  operator $Q$ is the covariance operator of $L$ considered as a cylindrical process in $U$, cf.~Remark~\\ref{rem:Q} below. As a consequence of Douglas' theorem as stated in \\cite[Appendix A.4]{PesZab07}, compare also \\cite[Corollary C.0.6]{PreRoeck07}, the reproducing kernel Hilbert space of $L$ has the alternative representation\n\\begin{equation*}\n(U_0,\\langle\\arg,\\arg\\rangle_{U_0})=\\big(Q^{1\/2}(U),\\langle Q^{-1\/2}\\arg,Q^{-1\/2}\\arg\\rangle_U\\big).\n\\end{equation*}\n\n\\begin{remark}\\label{rem:Q}\nSuppose w.l.o.g.\\ that $U$ is dense in $U_1$, identify $U$ and $U^*$ via the Riesz isomorphism, and consider the Gelfand triple $U_1^*\\subset U^*\\equiv U\\subset U_1$. Then it is not difficult to see that\n\\[\\bE\\langle L(t),x\\rangle\\langle L(t),y\\rangle=t\\langle Qx,y\\rangle_U,\\quad t\\geq0,\\;x,y\\in U_1^*,\\]\nwhere $\\langle\\arg,\\arg\\rangle:U_1\\times U_1^*\\to\\bR$ is the canonical dual pairing; compare \\cite[Proposition 7.7]{PesZab07}.\nThe unique continuous extensions of the linear mappings $U_1^*\\ni x\\mapsto \\langle L(t),x\\rangle\\in L^2(\\bP)$, $t\\geq0$, to the larger space $U^*$ determine a $2$-cylindrical $U$-process in the sense of \\cite{MetPel80}, compare also \\cite{AppRie10}, \\cite{Rie12}, \\cite{Rie14}.\n\\end{remark}\n\n\\begin{remark}\\label{rem:approxL}\nUnlike in the case of a mean-zero (cylindrical) $Q$-Wiener process in $U$, the covariance operators $Q\\in\\ensuremath{\\mathscr L}(U)$ and $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ do \\emph{not} determine the distribution of the L\\'{e}vy process $L$, but the jump intensity measure $\\nu$ does so according to \\eqref{eq:charFunctionL}. Note that the law of a general L\\'{e}vy process is determined by its characteristics (L\\'{e}vy triplet), cf.~\\cite[Definition~4.28]{PesZab07}, and that the characteristics of $L$ are $(-\\int_{\\{\\nnrm{y}{U_1}\\geq1\\}}y\\,\\nu(\\ensuremath{\\mathrm{d}} y),0,\\nu)$. Nevertheless, the operator $Q$ in $\\eqref{eq:defQ}$ will play an important role in our error analysis.\nLet us briefly make the connection of our setting to the construction of a cylindrical $Q$-Wiener process in $U$ as described in \\cite{DPZ92}, \\cite{PreRoeck07}. To this end, let $(f_k)_{k\\in\\bN}$ be an orthonormal basis of $U_1$ consisting of eigenvectors of $Q_1$ with eigenvalues $(\\lambda_k)_{k\\in\\bN}$ and consider the orthonormal basis $(e_k)_{k\\in\\bN}$ of $U_0$ given by $e_k:=\\lambda_k^{1\/2}f_k$. To simplify notation we suppose for the moment that all eigenvalues $\\lambda_k$ of $Q_1$ are strictly positive. Then, compare \\cite[Section~4.8]{PesZab07}, the real-valued L\\'{e}vy processes $L_k=(L_k(t))_{t\\geq0}$, $k\\in\\bN$, given by\n\\[L_k(t):=\\lambda_k^{-1\/2}\\langle L(t),f_k\\rangle_{U_1}\\]\nare uncorrelated, i.e., $\\bE L_k(t)L_j(s)=0$ if $k\\neq j$, they satisfy $\\bE(L_k^2(t))=t$, and we have\n\\begin{equation}\\label{eq:LPexpansion}\nL(t)=\\sum_{k\\in\\bN}L_k(t)e_k.\n\\end{equation}\nThe infinite sum in \\eqref{eq:LPexpansion} converges for all finite $T>0$ in the space $\\cM^2_T(U_1)$ of  c\\`{a}dl\\`{a}g square-integrable $U_1$-valued $(\\cF_t)$-martingales $M=(M(t))_{t\\in[0,T]}$ with norm $\\nnrm{M}{\\cM^2_T(U_1)}=(\\bE\\nnrm{M(T)}{U_1}^2)^{1\/2}$. In contrast to the Gaussian case, where uncorrelated coordinates are always independent, the coordinate processes $L_k$, $k\\in\\bN$, are in general only uncorrelated but \\emph{not} independent.\n\nConversely, suppose that we are given an arbitrary symmetric and nonnegative operator $Q\\in\\ensuremath{\\mathscr L}(U)$, an orthonormal basis $(e_k)_{k\\in\\bN}$ of $U_0=Q^{1\/2}(U)$, and a family $L_k$, $k\\in\\bN$, of real-valued L\\'{e}vy processes on $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ that satisfy the following conditions:\n\\begin{itemize}\n\\item Each $L_k$ is $(\\mathcal F_t)$-adapted and for $t,h\\geq0$ the increment $L_k(t+h)-L_k(t)$ is independent of $\\cF_t$;\n\\item each $L_k$ is square-integrable with $\\bE L_k(t)=0$ and $\\bE(L_k^2(t))=t$;\n\\item the processes $L_k$, $k\\in\\bN$, are uncorrelated;\n\\item for all $n\\in\\bN$ the $\\bR^n$-valued process $((L_1(t),\\ldots,L_n(t))^\\top)_{t\\geq 0}$ is a L\\'{e}vy process;\n\\item the Gaussian part of each $L_k$ is zero.\n\\end{itemize}\nThen, if $U_1$ is a Hilbert space containing $U$ such that the natural embedding of $U_0=Q^{1\/2}(U)$ into $U_1$ is Hilbert-Schmidt, the infinite sum in \\eqref{eq:LPexpansion} converges in $\\cM^2_T(U_1)$ and defines a L\\'{e}vy process $L$ with reproducing kernel Hilbert space $U_0$ that fits into our setting.\n\\end{remark}\n\nWe end this subsection with some examples of L\\'{e}vy processes $L$. We suppose that all processes are defined relative to the stochastic basis $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ and that their increments on time intervals $[t,t+h]$ are independent of $\\cF_t$.\n\n\\begin{example}(\\emph{Subordinate cylindrical $\\widetilde Q$-Wiener process})\nLet $W=(W(t))_{t\\geq0}$ be a cylindrical $\\widetilde Q$-Wiener process in $U$ in the sense of \\cite[Section 2.5.1]{PreRoeck07}, where $\\widetilde Q\\in\\ensuremath{\\mathscr L}(U)$ is a given nonnegative and symmetric operator. Assume that $W$ takes values in a possibly larger Hilbert space $U_1\\supset U$ such that the natural embedding of $U$ into $U_1$ is continuous. Let $\\widetilde Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ be the covariance operator of $W$ considered as a Wiener process in $U_1$, i.e., $\\bE\\langle W(t),x\\rangle_{U_1}\\langle W(s),y\\rangle_{U_1}=\\min(s,t)\\langle \\widetilde Q_1x,y\\rangle_{U_1}$ for $x,\\,y\\in U_1$, $s,t\\geq0$. Let $Z=(Z(t))_{t\\geq0}$ be a subordinator, i.e., a real-valued increasing L\\'{e}vy process in the sense of \\cite[Definition 21.4]{Sato13}, \\cite{SSV10}. Assume that $W$ and $Z$ are independent, that the drift of $Z$ is zero, and that the jump intensity measure  $\\rho$ of $Z$ satisfies\n\\begin{equation}\\label{eq:measureSubordinator}\n\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)<\\infty.\n\\end{equation}\nThe latter is equivalent to assuming that $Z$ has first moments, $\\bE|Z(t)|<\\infty$. According to \\cite[Remark~21.6]{Sato13}, the Laplace tranform of $Z(t)$ is given by\n\\begin{equation}\\label{eq:LaplaceSubordinator}\n\\bE(e^{-r Z(t)})=\\exp\\big(-t\\int_0^\\infty(1-e^{-rs})\\rho(\\ensuremath{\\mathrm{d}} s)\\big),\\quad r\\geq0.\n\\end{equation}\nIn this situation, subordinate cylindrical Brownian motion\n\\begin{equation*}\\label{eq:subWP}\nL(t):=W(Z(t)),\\quad t\\geq0,\n\\end{equation*}\ndefines a $U_1$-valued L\\'{e}vy process $L=(L(t))_{t\\geq0}$ that fits into the general framework described above. Indeed, $L$ has stationary and independent increments. Moreover, the independence of $W$ and $Z$, the identity $\\bE e^{i\\langle x,W(s)\\rangle_{U_1}}=e^{-s\\frac12 \\langle \\widetilde Q_1x,x\\rangle_{U_1}}$, Eq.~\\eqref{eq:LaplaceSubordinator} and the symmetry of the distribution $\\bP_{W(1)}=N(0,Q_1)$ imply that characteristic function of $L(t)$ is given by\n\\begin{align*}\n\\bE e^{i\\langle x,L(t)\\rangle_{U_1}}\n&=\\int_0^\\infty e^{-s\\frac12 \\langle\\widetilde Q_1x,x\\rangle_{U_1}}\\,\\bP_{Z(t)}(\\ensuremath{\\mathrm{d}} s)\\\\\n&=\\exp\\Big[-t\\int_0^\\infty(1-e^{-\\frac12\\langle\\widetilde Q_1x,x\\rangle_{U_1}s})\\rho(\\ensuremath{\\mathrm{d}} s)\\Big]\\\\\n&=\\exp\\Big[-t\\int_0^\\infty\\int_{U_1}(1-e^{i\\langle x,\\sqrt s y\\rangle_{U_1}}+i\\langle x,\\sqrt s y\\rangle_{U_1})\\bP_{W(1)}(\\ensuremath{\\mathrm{d}} y)\\rho(\\ensuremath{\\mathrm{d}} s)\\Big].\n\\end{align*}\nAs a consequence, \\eqref{eq:charFunctionL} holds with\n\\begin{equation}\\label{eq:subWPnu}\n\\nu=(\\bP_{W(1)}\\otimes\\rho)\\circ\\kappa^{-1},\n\\end{equation}\nwhere $\\kappa:U_1\\times(0,\\infty)\\to U_1$ is defined by $\\kappa(y,s)=\\sqrt s y$; compare \\cite[Lemma 4.8]{Rie14}. (Note that, by the scaling property of $W$, \\eqref{eq:subWPnu} is equivalent to the standard formula $\\nu=\\int_0^\\infty\\bP_{W(s)}\\,\\rho(\\ensuremath{\\mathrm{d}} s)$, where the measure-valued integral is defined in a weak sense, cf.~\\cite[Section~30]{Sato13}). Moreover, \\eqref{eq:assnu} holds due to \\eqref{eq:measureSubordinator} as we have the equality\n$\\int_{U_1}\\nnrm{y}{U_1}^2\\nu(\\ensuremath{\\mathrm{d}} y)=\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)\\,\\bE(\\nnrm{W(1)}{U_1}^2)$ according to \\eqref{eq:subWPnu}.\nIt follows that $L$ is a $U_1$-valued, square-integrable, mean-zero L\\'{e}vy process with vanishing Gaussian part. It is also not difficult to show that the covariance operators $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ and $Q\\in\\ensuremath{\\mathscr L}(U)$ of $L$ in \\eqref{eq:Q1nu} and \\eqref{eq:defQ} are given by\n$Q_1=\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)\\,\\widetilde Q_1$ and $Q=\\int_0^\\infty s\\,\\rho(\\ensuremath{\\mathrm{d}} s)\\,\\widetilde Q$.\nSubordinate cylindrical Wiener processes have been considered, e.g., in \\cite{BrzZab10}.\n\\end{example}\n\n\\begin{example}(\\emph{Independent one-dimensional L\\'{e}vy processes})\\label{ex:1dlevy}\nLet $Q\\in\\ensuremath{\\mathscr L}(U)$ be symmetric, nonnegative and let $(e_k)_{k\\in\\bN}$ be an orthonormal basis of $U_0:=Q^{1\/2}(U)\\subset U$. Let $L_k=(L(t))_{k\\in\\bN}$, $k\\in\\bN$, be independent real-valued square-integrable L\\'{e}vy processes with vanishing Gaussian part and $\\bE L_k(t)=0$, $\\bE(L_k^2(t))=t$. Let $U_1\\supset U$ be another Hilbert space such that the natural embedding of $U_0$ into $U_1$ is a Hilbert-Schmidt operator. Then, the series \\eqref{eq:LPexpansion} converges for all $T\\in(0,\\infty)$ in the space $\\cM^2_T(U_1)$ and defines a L\\'{e}vy process $L=(L(t))_{t\\geq0}$ satisfiying \\eqref{eq:assnu} and \\eqref{eq:charFunctionL} with jump intensity measure\n\\[\\nu=\\sum_{k\\in\\bN}\\nu_k\\circ\\pi_k^{-1},\\]\nwhere $\\nu_k$ is the L\\'{e}vy measure of $L_k$ and $\\pi_k:\\bR\\to U_1$ is defined by $\\pi_k(\\xi):=\\xi e_k$; compare \\cite[Section 4.8.1]{PesZab07}.\n\\end{example}\n\n\\begin{example}(\\emph{Impulsive cylindrical process})\nLet $\\mu$ be a L\\'{e}vy measure on $\\bR$ such that $\\int_\\bR\\sigma^2\\mu(\\ensuremath{\\mathrm{d}}\\sigma)<\\infty$. Let $\\cO\\subset\\bR^d$ be a bounded domain and $Z=(Z(t))_{t\\geq0}$ an impulsive cylindrical process on $U:=L^2(\\cO)=L^2(\\cO,\\cB(\\cO),\\lambda^d)$ with jump size intensity $\\mu$ in the sense of \\cite[Definition 7.23]{PesZab07}. Here, $\\lambda^d$ denotes $d$-dimensional Lebesgue measure.  The process $Z$ is a measure-valued process defined, informally, by $Z(t,\\ensuremath{\\mathrm{d}}\\xi)=\\int_0^t\\int_\\bR\\sigma\\hat\\pi(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}}\\xi,\\ensuremath{\\mathrm{d}} \\sigma)$, where $\\hat\\pi$ is a compensated Poisson random measure on $[0,\\infty)\\times\\cO\\times\\bR$ with reference measure $\\lambda^1\\otimes\\lambda^d\\otimes\\mu$; see \\cite[Section 7.2]{PesZab07} for details. Let $\\widetilde Q\\in\\ensuremath{\\mathscr L}(U)$ be symmetric and nonnegative, $(b_k)_{k\\in\\bN}$ an orthonormal basis of $U$, and $U_1\\supset U$ a Hilbert space such that the natural embedding of $U_0=\\tilde Q^{1\/2}(U)\\subset U$ into $U_1$ is Hilbert-Schmidt. Then the series\n\\begin{equation}\\label{eq:imulsiveProcess}\nL(t):=\\widetilde Q^{\\frac12} Z(t):=\\sum_{k\\in\\bN}\\int_0^t\\int_\\cO\\int_\\bR\\sigma b_k(\\xi)\\hat\\pi(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}}\\xi,\\ensuremath{\\mathrm{d}} \\sigma)\\widetilde Q^{\\frac12}b_k,\\quad t\\geq0,\n\\end{equation}\nconverges for all $T\\in(0,\\infty)$ in $\\cM^2_T(U_1)$ and defines a L\\'{e}vy process that fits into our general framework with $Q=\\int_\\bR\\sigma^2\\mu(\\ensuremath{\\mathrm{d}}\\sigma)\\widetilde Q$ and\n\\[\\nu=(\\lambda^d\\otimes\\mu)\\circ \\phi^{-1},\\]\nwhere $\\phi\\in L^2(\\cO\\times\\bR,\\lambda^d\\otimes\\mu;U_1)$ is defined by $\\phi(\\xi,\\sigma)=\\sum_{n\\in\\bN}\\sigma b_k(\\xi)\\widetilde Q^{\\frac12}b_k$ (convergence in $L^2(\\cO\\times\\bR,\\lambda^d\\otimes\\mu;U_1)$).\nIn \\cite{LinSch13} we considered the weak approximation of the stochastic heat equation driven by an impulsive process of the form \\eqref{eq:imulsiveProcess}. The results in Section~\\ref{subsec:she} of the present article improve the results of \\cite{LinSch13} in several aspects.\n\\end{example}\n\n\\subsection{Linear stochastic evolution equations with additive noise}\n\\label{sec:LSEE}\n\nWe are mainly interested in equations of the type \\eqref{eq:mainEq}, where $A:D(A)\\subset H\\to H$ is an unbounded linear operator such that $-A$ is the generator of a strongly continuous semigroup $(E(t))_{t\\geq0}\\subset\\ensuremath{\\mathscr L}(H)$, $B\\in\\ensuremath{\\mathscr L}(U,H)$, $L=(L(t))_{t\\geq0}$ is a square-integrable L\\'{e}vy process with reproducing kernel Hilbert space $U_0\\subset U$ as described in Subsection~\\ref{sec:DrivingProcess}, and $X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$. It is well known that if\n\\begin{equation}\\label{eq:assEB1}\n\\int_0^T \\nnrm{E(t)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} t <\\infty\n\\end{equation}\nfor some (and hence for all) $T>0$, then there exists a unique weak solution $X=(X(t))_{t\\geq0}$ to \\eqref{eq:mainEq} which is given by the variation-of-constants formula \\eqref{eq:X}, see, e.g., \\cite[Chapter~9]{PesZab07}.\nSimilarly, if $(\\tilde E(t))_{t\\in[0,T]}\\subset\\ensuremath{\\mathscr L}(H)$\nis given by some approximation scheme such that $t\\mapsto\\tilde E(t)B$ is a measurable mapping from $[0,T]$ to $\\ensuremath{\\mathscr L_2}(U_0,H)$, then the condition\n\\begin{equation}\\label{eq:assEBtilde1}\n\\int_0^T \\nnrm{\\tilde E(t)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} t <\\infty\n\\end{equation}\nensures that the approximation $\\tilde X=(\\tilde X(t))_{t\\in[0,T]}$ of $(X(t))_{t\\in[0,T]}$ in \\eqref{eq:Xtilde} exists as a square-integrable $H$-valued process. We refer to \\cite[Chapter~8]{PesZab07} for details on the construction and properties of the stochastic integral w.r.t.\\ Hilbert space-valued L\\'{e}vy processes.\n\nIt turns out that our general error representation formula for the weak error $e(T)$ in \\eqref{eq:defWeakError} does not require the semigroup property of the strongly continuous family of operators $(E(t))_{t\\geq0}$. This paves the way for analysing a more general class of L\\'{e}vy-driven linear stochastic evolution equations, including for example stochastic Volterra-type equations as considered in \\cite{KovPri14a}, \\cite{KovPri14b} for the Gaussian case, see Subsection \\ref{subsec:sve} for an example.  For such equations, the weak solution still has the form  \\eqref{eq:X} but the solution operator family $(E(t))_{t\\geq0}\\subset\\ensuremath{\\mathscr L}(H)$ is not a semigroup anymore.  Therefore, we weaken our abstract assumptions and summarize them as follows.\n\n\\begin{assumption} \\label{ass:abstractSetting}\nWe will use the following assumptions:\n\\begin{compactenum}[(i)]\n\\item $H$, $U$ and $U_1$ are real and separable Hilbert spaces;\n\\item\n$L=(L(t))_{t\\geq0}$ is a $U_1$-valued L\\'{e}vy process on $(\\Omega,\\cF,(\\cF_t)_{t\\geq0},\\bP)$ with zero mean and finite second moments and reproducing kernel Hilbert space $U_0$ such that $U_0\\subset U\\subset U_1$ as described in Subsection~\\ref{sec:DrivingProcess};\n\\item\n$X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$;\n\\item\n$B\\in\\ensuremath{\\mathscr L}(U,H)$ and $(E(t))_{t\\in[0,T]}\\subset \\ensuremath{\\mathscr L}(H)$ is a strongly continuous family of linear operators such that \\eqref{eq:assEB1} holds;\n\\item\nfor all $\\epsilon>0$ there exists $\\Phi_\\epsilon\\in\\ensuremath{\\mathscr L_2}(U_0,H)$ and $C_\\epsilon>0$ such that\n\\[\\nnrm{E(t)Bx}{H}\\leq \\nnrm{\\Phi_\\epsilon x}H,\\quad (t,x)\\in[\\epsilon,T]\\times U_0;\\]\n\\item\n$(\\tilde E(t))_{t\\in[0,T]}\\subset\\ensuremath{\\mathscr L}(H)$ is a family of linear operators such that $t\\mapsto\\tilde E(t)B$ is a measurable mapping from $[0,T]$ to $\\ensuremath{\\mathscr L_2}(U_0,H)$ and \\eqref{eq:assEBtilde1} holds;\n\\item\n$X=(X(t))_{t\\in[0,T]}$ and $\\tilde X=(\\tilde X(t))_{t\\in[0,T]}$ are $H$-valued stochastic processes given by \\eqref{eq:X} and \\eqref{eq:Xtilde}.\n\\end{compactenum}\n\\end{assumption}\n\n\\begin{remark}\nIf $(E(t))_{t\\geq0}$ is an operator semigroup, then \\ref{ass:abstractSetting}(v) is a consequence of \\ref{ass:abstractSetting}(iv). Indeed, if \\eqref{eq:assEB1} holds then $E(t_0)B\\in \\ensuremath{\\mathscr L_2}(U_0,H)$ for some $t_0\\in (0,\\epsilon)$ and hence\n\\[\\nnrm{E(t)Bx}{H}\\le \\|E(t_0)Bx\\|_H\\sup_{t\\in [t_0,T]}\\|E(t-t_0)\\|_{\\ensuremath{\\mathscr L}(H)},\\quad(t,x)\\in[\\epsilon,T]\\times U_0.\\]\nHence, one may take $\\Phi_{\\epsilon}:=cE(t_0)B$ where $c:=\\sup_{t\\in [t_0,T]}\\|E(t-t_0)\\|_{\\ensuremath{\\mathscr L}(H)}$.\n\\end{remark}\n\nTo fix notation, let us briefly recall the It\\^{o} isometry for stochastic integrals w.r.t.\\ $L$. It has the same form as the It\\^{o} isometry for stochastic integrals w.r.t.\\ Hilbert space-valued Wiener processes.\nWe set $\\Omega_T:=\\Omega\\times[0,T]$ and $\\bP_T:=\\bP\\otimes\\lambda$, where $\\lambda$ is Lebesgue measure on $[0,T]$. The predictable $\\sigma$-algebra on $\\Omega_T$ w.r.t.\\ $(\\cF_t)_{t\\in[0,T]}$ is denoted by $\\cP_T$. For operator-valued processes $\\Phi=(\\Phi(t))_{t\\in[0,T]}$ in\n\\[L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H)):=L^2(\\Omega_T,\\cP_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H)),\\]\nwe have\n\\begin{equation}\\label{eq:ItoIsom}\n\\bE\\Big(\\gnnrm{\\int_0^t\\Phi(s)\\ensuremath{\\mathrm{d}} L(s)}H^2\\Big)=\\bE\\int_0^t\\nnrm{\\Phi(s)}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} s,\\quad t\\in[0,T],\n\\end{equation}\nand the integral process $(\\int_0^t\\Phi(s)\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$ belongs to the space $\\cM^2_T(H)$ of  c\\`{a}dl\\`{a}g square-integrable $H$-valued $(\\cF_t)$-martingales. The usual norm in $\\cM^2_T(H)$ is defined by $\\nnrm{M}{\\cM^2_T(H)}=(\\bE\\nnrm{M(T)}{H}^2)^{1\/2}$, $M=(M(t))_{t\\in[0,T]}\\in\\cM^2_T(H)$. Note, however, that the integral processes given by the stochastic integrals in \\eqref{eq:X} and \\eqref{eq:Xtilde} are in general \\emph{not} martingales since the (deterministic) operator-valued integrands also depend on $t$.\n\nWe also recall\nthe definition and some properties of Hilbert-Schmidt operators,\ncf.~\\cite[Chapter~6]{Wei80}. Let $\\cH$ and $\\cG$ be real and separable Hilbert spaces. A linear and bounded operator $C\\in\\ensuremath{\\mathscr L}(\\cH,\\cG)$ belongs to the space $\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ of Hilbert-Schmidt operators if\n\\[\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}:=\\Big(\\sum_{k\\in\\bN}\\nnrm{Ch_k}{\\cG}^2\\Big)^{1\/2}<\\infty\\]\nfor some (and hence for every) orthonormal basis $(h_k)_{k\\in\\bN}$ of $\\cH$. If $C\\in\\ensuremath{\\mathscr L}(\\cH,\\cG)$ and $C^*\\in\\ensuremath{\\mathscr L}(\\cG,\\cH)$ is the adjoint operator, then $C\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ if and only if $C^*\\in\\ensuremath{\\mathscr L_2}(\\cG,\\cH)$ and one has\n\\begin{equation}\\label{eq:propHSadjoint}\n\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}=\\nnrm{C^*}{\\ensuremath{\\mathscr L_2}(\\cG,\\cH)}.\n\\end{equation}\nAlso, if $C\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$, $D\\in\\ensuremath{\\mathscr L}(\\cH)$ and $F\\in\\ensuremath{\\mathscr L}(\\cG)$, then obviously $CD\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$, $FC\\in\\ensuremath{\\mathscr L_2}(\\cH,\\cG)$ and\n\\begin{equation}\\label{eq:estHS}\n\\nnrm{CD}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}\\leq\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}\\nnrm{D}{\\ensuremath{\\mathscr L}(\\cH)},\\quad\n\\nnrm{FC}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}\\leq\\nnrm{F}{\\ensuremath{\\mathscr L}(\\cG)}\\nnrm{C}{\\ensuremath{\\mathscr L_2}(\\cH,\\cG)}.\n\\end{equation}\nIn particular, in our setting we have $\\ensuremath{\\mathscr L}(U_1,H)\\subset\\ensuremath{\\mathscr L_2}(U_0,H)$ since\n\\[\\nnrm{C}{\\ensuremath{\\mathscr L_2}(U_0,H)}=\\nnrm{CQ_1^{1\/2}}{\\ensuremath{\\mathscr L_2}(U_1,H)}\\leq\\nnrm{C}{\\ensuremath{\\mathscr L}(U_1,H)}\\nnrm{Q_1^{1\/2}}{\\ensuremath{\\mathscr L_2}(U_1)}\\]\nfor all $C\\in\\ensuremath{\\mathscr L}(U_1,H)$ and $\\nnrm{Q_1^{1\/2}}{\\ensuremath{\\mathscr L_2}(U_1)}=\\text{Tr\\,}Q_1=\\nnrm{Q_1}{\\ensuremath{\\mathscr L_1}{(U_1)}}<\\infty$.\n\n\\section{An error representation formula}\n\\label{sec:An error representation formula}\n\nIn this section, we state and prove a general representation formula for the weak approximation error $e(T)$ in \\eqref{eq:defWeakError} within the abstract setting described above.\n\n\\subsection{Formulation of the result}\n\nFor the formulation and the proof of the error representation formula, we introduce auxiliary drift-free It\\^{o} processes $Y=(Y(t))_{t\\in[0,T]}$ and $\\tilde Y=(\\tilde Y(t))_{t\\in[0,T]}$ such that\n\\begin{equation*}\nX(T)=Y(T),\\quad \\tilde X(T)=\\tilde Y(T).\n\\end{equation*}\nThe processes $Y$ and $\\tilde Y$ are constructed by applying to $X$ and $\\tilde X$ the deterministic operator-valued processes $(E(T-t))_{t\\in[0,T]}$ and $(\\tilde E(T-t))_{t\\in[0,T]}$. That is, we set\n\\begin{equation}\\label{eq:defY}\nY(t):=E(T)X_0+\\int_0^tE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s), \\quad t\\in[0,T],\n\\end{equation}\nand\n\\begin{equation}\\label{eq:defYtilde}\n\\tilde Y(t):=\\tilde E(T)X_0+\\int_0^t\\tilde E(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s), \\quad t\\in[0,T].\n\\end{equation}\n\nMoreover, we consider the auxiliary problem\n\\[\\ensuremath{\\mathrm{d}} Z(t)=E(T-t)B\\,\\ensuremath{\\mathrm{d}} L(t),\\quad t\\in[\\tau,T];\\qquad Z(\\tau)=\\xi,\\]\nwhere $\\tau\\in[0,T)$ and $\\xi$ is an $H$-valued $\\cF_\\tau$-measurable random variable. Its solution is given by\n\\begin{equation}\\label{eq:defZttauxi}\nZ(t,\\tau,\\xi):=\\xi+\\int_{\\tau}^t E(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s),\\quad t\\in[\\tau,T],\n\\end{equation}\nand we use it to define for $G\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$ a function $u:[0,T]\\times H\\to\\bR$ by\n\\begin{equation}\\label{eq:defuxt}\nu(t,x):=\\bE\\,G(Z(T,t,x)),\\quad (t,x)\\in [0,T]\\times H.\n\\end{equation}\nNote that the boundedness of $G''$ implies quadratic and linear growth of $G$ and $G'$, respectively. That is,\n\\begin{equation}\\label{eq:growthGG'}\n|G(x)|\\leq C(1+\\nnrm{x}H^2),\\quad G'(x)\\leq C(1+\\nnrm{x}H)\n\\end{equation}\nfor all $x\\in H$ and a constant $C\\in(0,\\infty)$ that does not depend on $x$.\nIt is also not difficult to see that $u$\nis twice Fr\\'{e}chet differentiable w.r.t. $x$ and we have\n\\begin{equation}\\label{eq:u_xu_xx}\nu_x(t,x)=\\bE\\,G'(Z(T,t,x)),\\quad u_{xx}(t,x)=\\bE\\,G''(Z(T,t,x)).\n\\end{equation}\nAll expectations appearing in \\eqref{eq:defuxt} and \\eqref{eq:u_xu_xx} make sense due to \\eqref{eq:growthGG'}, the assumption $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$, \\eqref{eq:assEB1} and It\\^{o}'s isometry \\eqref{eq:ItoIsom}.\n\nBefore stating the representation formula, we show in the following lemma how operators in $\\ensuremath{\\mathscr L_2}(U_0,H)$ can be identified with functions in\n\\begin{equation*}\nL^2(U_1,\\nu;H):=L^2(U_1,\\mathcal B(U_1),\\nu;H)\n\\end{equation*}\nand how processes in $L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ can be identified with elements in\n\\begin{equation*}\nL^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H):=L^2(\\Omega_T\\times U_1,\\cP_T\\otimes \\cB(U_1),\\bP_T\\otimes\\nu;H).\n\\end{equation*}\nThese identifications will be used implicitly throughout this article, see Remark~\\ref{not:Phix} below. They also lead to a generic identification of integrals w.r.t.\\ (cylindrical) Hilbert space-valued L\\'{e}vy processes of jump type and integrals w.r.t.\\ the associated Poisson random measures, cf.\\ Appendix~\\ref{sec:PRM+SI}.\n\n\n\\begin{lemma}\\label{lem:integrandIsomorphism}\nLet $(f_k)_{k\\in\\bN}\\subset U_0$ be an orthonormal basis of $U_1$ consisting of eigenvectors of the covariance operator $Q_1\\in\\ensuremath{\\mathscr L_1}(U_1)$ of $L$ and let $(\\lambda_k)_{k\\in\\bN}\\subset[0,\\infty)$ be the corresponding sequence of eigenvalues\n\n\\textbf{(i)}\nGiven $\\Phi\\in\\ensuremath{\\mathscr L_2}(U_0,H)$, the series\n\\[\\iota(\\Phi):=\\sum_{k\\in\\bN,\\lambda_k\\neq 0}\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k\\]\nconverges in $L^2(U_1,\\nu; H)$.\n\nThe linear mapping\n\\[\\iota:\\ensuremath{\\mathscr L_2}(U_0,H)\\to L^2(U_1,\\nu;H),\\;\\Phi\\mapsto \\iota(\\Phi)\\]\nis an isometric embedding.\n\n\\textbf{(ii)}\nGiven $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$, the series\n\\[\\kappa(\\Phi):=\\sum_{k\\in\\bN,\\lambda_k\\neq0}\\langle\\arg,f_k\\rangle_{U_1}\\Phi(\\arg) f_k\\]\nconverges in $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$.\nThe linear mapping\n\\[\\kappa:L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))\\to L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H),\\;\\Phi\\mapsto \\kappa(\\Phi)\\]\nis an isometric embedding.  For $\\bP_T$-almost all $(\\omega,t)\\in\\Omega_T$ we have\n$\\kappa(\\Phi)(\\omega,t,\\arg)=\\iota(\\Phi(\\omega,t))$\nin $L^2(U_1,\\nu;H)$, where $\\iota$ is the embedding from \\textup{(i)}.\n\\end{lemma}\n\n\\begin{proof\n\\textbf{(i)}\nW.l.o.g.\\ all eigenvalues $\\lambda_k$ are strictly positive. Let $(e_k)_{k\\in\\bN}$ be the orthonormal basis of $U_0$ given by $e_k:=\\lambda_k^{1\/2}f_k$. For $m,n\\in\\bN$ with $m\\leq n$ we have\n\\begin{align*}\n\\sgnnrm{\\sum_{k=m}^n\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k}{L^2(U_1,\\nu;H)}^2\n&=\\int_{U_1}\\sgnnrm{\\sum_{k=m}^n\\langle x,f_k\\rangle_{U_1}\\Phi f_k}{H}^2\\nu(\\ensuremath{\\mathrm{d}} x)\\\\\n&= \\sum_{j,k=m}^n\\lambda_j^{-1\/2}\\lambda_k^{-1\/2}\\int_{U_1}\\langle x,f_j\\rangle_{U_1}\\langle x,f_k\\rangle_{U_1}\\nu(\\ensuremath{\\mathrm{d}} x)\\,\\langle\\Phi e_j,\\Phi e_k\\rangle_H\\\\\n&=\\sum_{k=m}^n\\nnrm{\\Phi e_k}H^2;\n\\end{align*}\nin the last step we used \\eqref{eq:Q1nu}. Since $\\sum_{k\\in\\bN}\\nnrm{\\Phi e_k}H^2=\\nnrm{\\Phi}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2<\\infty$, this shows that the partial sums $\\sum_{k=1}^n\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k$, $n\\in\\bN$, are a Cauchy sequence in $L^2(U_1,\\nu;H)$ and\n\\[\\sgnnrm{\\sum_{k=1}^\\infty\\langle\\arg,f_k\\rangle_{U_1}\\Phi f_k}{L^2(U_1,\\nu;H)}=\\nnrm{\\Phi}{\\ensuremath{\\mathscr L_2}(U_0,H)}.\\]\n\n\n\\textbf{(ii)} The first two assertions can be shown as in the proof of (i). The last assertion is due the fact that the iterated integral\n\\[\\int_\\Omega\\int_0^T\\int_{U_1}\\nnrm{\\iota(\\Phi(\\omega,t))(x)-\\kappa(\\Phi)(\\omega,t,x)}H^2\\,\\nu(\\ensuremath{\\mathrm{d}} x)\\,\\ensuremath{\\mathrm{d}} t\\,\\bP(\\ensuremath{\\mathrm{d}}\\omega)\\]\nequals zero, which follows from an approximation argument.\n\\end{proof}\n\n\\begin{remark}\\label{not:Phix}\nFrom now on we will\nidentify operators $\\Phi\\in\\ensuremath{\\mathscr L_2}(U_0,H)$ with the corresponding mappings $\\iota(\\Phi)\\in L^2(U_1,\\nu;H)$ and write\n\\[\\Phi x=\\iota(\\Phi)(x),\\quad x\\in U_1.\\]\nAnalogously, we identify processes $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ with the corresponding mappings $\\kappa(\\Phi)\\in L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$ and write\n\\[\\Phi(\\omega,t)x=\\kappa(\\Phi)(\\omega,t,x),\\quad (\\omega,t,x)\\in \\Omega_T\\times U_1.\\]\nFor processes $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ both identifications\nare compatible $\\bP\\otimes\\lambda$-almost everywhere on $\\Omega_T$ in the sense that we have $\\kappa(\\Phi)(\\omega,t,\\arg)=\\iota(\\Phi(\\omega,t))$\nin $L^2(U_1,\\nu;H)$ for $\\bP\\otimes\\lambda$-almost all $(\\omega,t)\\in\\Omega_T$.\n\\end{remark}\n\nHere is the main result of this section.\n\\begin{theorem}\\label{thm:errorRep}\nLet Assumption~\\ref{ass:abstractSetting} hold and $G\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$.\nThen, for the process $(\\tilde Y(t))_{t\\in[0,T]}$ from \\eqref{eq:defYtilde} and the function $u:[0,T]\\times H\\to\\bR$ from \\eqref{eq:defuxt} it holds that\n\\begin{equation}\\label{eq:errorRep1}\n\\begin{aligned}\n\\bE\\int_0^T\\int_{U_1}&\\Big|u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&-\\big\\langle u_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\;<\\;\\infty.\n\\end{aligned}\n\\end{equation}\nThe weak error $e(T)$ in \\eqref{eq:defWeakError} has the representation\n\\begin{equation}\\label{eq:errorRep2}\n\\begin{aligned}\ne(T)&=\\bE\\big\\{u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\\\\n&\\quad +\\bE\\int_0^T\\int_{U_1}\\Big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\hspace{3.5cm}-\\big\\langle u_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\nThe terms $E(T-t)By$ and $\\tilde E(T-t) By$ appearing in \\eqref{eq:errorRep1} and \\eqref{eq:errorRep2} are defined for $\\lambda\\otimes\\nu$-almost all $(t,y)\\in[0,T]\\times U_1$. This follows from \\eqref{eq:assEB1}, \\eqref{eq:assEBtilde1}, Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix},\n\\end{remark}\n\nWe will prove Theorem~\\ref{thm:errorRep} in the next subsection. Let us briefly record an alternative representation of $e(T)$ which follows from Taylor's formula.\nIt will be the starting point for our error estimates in Sections \\ref{sec:Heat} and \\ref{sec:Wave}. For $t\\in [0,T]$, $\\theta\\in[0,1]$ and $y\\in U_1$ set\n\\begin{align*}\nF(t)&:=\\tilde E(t) B-E(t)B,\\\\\n\\Psi_1(t,\\theta,y)&:=(1-\\theta)\\Big\\langle u_{xx}\\big(t,\\tilde Y(t)+ E(T-t)By+\\theta F(T-t)y\\big)F(T-t)y\\,,\\,F(T-t)y\\Big\\rangle_H,\\\\\n\\Psi_2(t,\\theta,y)&:=\\Big\\langle u_{xx}\\big(t,\\tilde Y(t)+\\theta E(T-t)By\\big)E(T-t)By\\,,\\,F(T-t)y\\Big\\rangle_H.\n\\end{align*}\n\n\\begin{corollary}\\label{cor:errorRep}\nIn the setting of Theorem~\\ref{thm:errorRep} we have\n\\begin{equation}\\label{eq:errorRep3}\n\\bE\\int_0^T\\int_{U_1}\\int_0^1\\big\\{|\\Psi_1(t,\\theta,y)|+|\\Psi_2(t,\\theta,y)|\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t<\\infty,\n\\end{equation}\nand the following alternative error representation holds:\n\\begin{equation}\\label{eq:errorRep4}\n\\begin{aligned}\ne(T)&=\\bE\\big\\{u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\\\\n&\\quad +\\bE\\int_0^T\\int_{U_1}\\int_0^1\\big\\{\\Psi_1(t,\\theta,y)+\\Psi_2(t,\\theta,y)\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation}\n\\end{corollary}\n\n\\begin{proof\nThe integrand of the iterated integral in \\eqref{eq:errorRep2} can be rewritten as\n\\begin{align*}\nu\\big(t,\\tilde Y(t)+&\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)-\\big\\langle u_x(t,\\tilde Y(t)),F(T-t)y\\big\\rangle_H\\\\\n&=\\Big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\quad-\\big\\langle u_x\\big(t,\\tilde Y(t)+E(T-t)By\\big),F(T-t)y\\big\\rangle_H\\Big\\}\\\\\n&\\quad+\\big\\langle u_x\\big(t,\\tilde Y(t)+E(T-t)By\\big)-u_x(t,\\tilde Y(t)),F(T-t)y\\big\\rangle_H\\\\\n&=\\int_0^1\\big\\{\\Psi_1(t,\\theta,y)+\\Psi_2(t,\\theta,y)\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta,\n\\end{align*}\nwhere the last step is due to Taylor's formula. By \\eqref{eq:errorRep1} we have\n\\[\\bE\\int_0^T\\int_{U_1}\\Big|\\int_0^1\\big\\{\\Psi_1(t,\\theta,y)+\\Psi_2(t,\\theta,y)\\big\\}\\,\\ensuremath{\\mathrm{d}}\\theta\\Big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t<\\infty.\\]\nThe stronger assertion \\eqref{eq:errorRep3} follows from the boundedness of $G'':H\\to\\ensuremath{\\mathscr L}(H)$, Lemma~\\ref{lem:integrandIsomorphism}, \\eqref{eq:assEB1} and \\eqref{eq:assEBtilde1}.\n\\end{proof}\n\n\n\n\\subsection{Proof of the error representation formula}\n\nIn this subsection, we give the proof of Theorem~\\ref{thm:errorRep}.\n\nFor $\\xi\\in L^2(\\Omega,\\cF_t,\\bP;H)$ we have\n\\begin{equation*}\\label{eq:Eu(t,xi)}\n\\bE\\big(G(Z(T,t,\\xi))\\big)\n=\\int_H\\int_HG(x+y)\\,\\bP_{\\int_t^TE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)}(\\ensuremath{\\mathrm{d}} y)\\bP_\\xi(\\ensuremath{\\mathrm{d}} x)\n=\\bE\\big(u(t,\\xi)\\big)\n\\end{equation*}\nby \\eqref{eq:defZttauxi}, \\eqref{eq:defuxt}, the independence of $\\int_t^TE(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)$ and $\\cF_t$, and Fubini's theorem.\nSince $X(T)=Y(T)$ and $\\tilde X(T)=\\tilde Y(T)$ it follows that\n\\begin{equation}\\label{eq:proofErrRep1}\n\\begin{aligned}\ne(T)&=\\bE\\big(G(\\tilde Y(T))-G(Y(T))\\big)\\\\\n&=\\bE\\big(G(Z(T,T,\\tilde Y(T)))-G(Z(T,0,Y(0)))\\big)\\\\\n&=\\bE\\big(u(T,\\tilde Y(T))-u(0,Y(0))\\big)\\\\\n&=\\bE\\big(u(0,\\tilde Y(0))-u(0,Y(0))\\big)+\\bE\\big(u(T,\\tilde Y(T))-u(0,\\tilde Y(0))\\big).\n\\end{aligned}\n\\end{equation}\nBy \\eqref{eq:defY} and \\eqref{eq:defYtilde}, the first term in the last line equals $\\bE(u(0,\\tilde E(T)X_0)-u(0,E(T)X_0))$.\n\nTo handle the second term in the last line of \\eqref{eq:proofErrRep1}, we first assume that $G:H\\to\\bR$ and $G_x:H\\to H$ are bounded, so that $G\\in C^2_{\\operatorname b}(H,\\bR)$. We will remove this restriction later on. We now want to apply It\\^{o}'s formula to the function $(t,x)\\mapsto u(t,x)$ and the martingale $\\tilde Y=(\\tilde Y(t))_{t\\in[0,T]}$. For this we need the following properties of $u$.\n\n\n\\begin{proposition}\\label{prop:backwardKolm}\nLet Assumption~\\ref{ass:abstractSetting} hold and $G\\in C_{\\operatorname b}^2(H,\\bR)$. The function $u:[0,T]\\times H\\to\\bR,\\;(t,x)\\mapsto u(t,x)$ defined in \\eqref{eq:defuxt} and its Fr\\'{e}chet partial derivatives $u_x$, $u_{xx}$ are continuous and bounded on $[0,T]\\times H$. The time derivative $u_t$ of $u$ exists on $[0,T)\\times H$ and is continuous.\nMoreover, for every $\\epsilon>0$ there exists some $C_\\epsilon<\\infty$ such that\n\\begin{equation}\\label{eq:backwardKolm0}\n\\int_{U_1} \\big|u\\big(t,x+E(T-t)By\\big)- u(t,x)-\\big\\langle u_x(t,x),E(T-t)By\\big\\rangle_H\\big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)<C_\\epsilon\n\\end{equation}\nfor all $t\\in[0,T-\\epsilon]$, and $u$ satisfies the backward Kolmogorov equation\n\\begin{equation}\\label{eq:backwardKolm1}\n\\left.\n\\begin{aligned}\nu_t(t,x)&=-\\int_{U_1} \\Big\\{u\\big(t,x+E(T-t)By\\big)-u(t,x)-\\big\\langle u_x(t,x),E(T-t)By\\big\\rangle_H\\Big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y),\\\\\n&\\hspace{2cm}(t,x)\\in [0,T)\\times H,\\\\\nu(T,x)&=G(x),\\quad x\\in H.\n\\end{aligned}\n\\;\\right\\}\n\\end{equation}\n\\end{proposition}\n\n\n\\begin{proof}\n\nWe begin with the continuity and boundedness of $u$, $u_x$ and $u_{xx}$. The boundedness is obvious by the definition \\eqref{eq:defuxt} of $u$ and by \\eqref{eq:u_xu_xx}. Pick $0\\leq s\\leq t\\leq T$, $x,y\\in H$. Using \\eqref{eq:defuxt}, Jensen's inequality, the mean value theorem, \\eqref{eq:defZttauxi} and It\\^{o}'s isometry,  we have\n\\begin{equation*}\n\\begin{aligned}\n|u(t,x)-u(s,y)|^2&\\leq\\bE\\big(|G(Z(T,t,x))-G(Z(T,s,y))|^2\\big)\\\\\n&\\leq \\sup_{x\\in H}\\nnrm{G'(x)}H^2\\,\\bE\\Big(\\gnnrm{x-y-\\int_s^t E(T-r)B\\,\\ensuremath{\\mathrm{d}} L(r)}H^2\\Big)\\\\\n&\\leq 2\\sup_{x\\in H}\\nnrm{G'(x)}H^2\\Big(\\nnrm{x-y}H^2+\\int_s^t\\nnrm{E(T-r)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\,\\ensuremath{\\mathrm{d}} r\\Big).\n\\end{aligned}\n\\end{equation*}\nThus, the continuity of $u$ follows from \\eqref{eq:assEB1} and the boundedness of $G'$.\nSince $u_x(t,x)=\\bE\\,G'(Z(T,t,x))$, the continuity of $u_x:[0,T]\\times H\\to H$ follows analogously from the boundedness of $G''$.\nTo show the continuity of $u_{xx}:[0,T]\\times H\\to\\ensuremath{\\mathscr L}(H)$, define $F\\in C_{\\operatorname b}(H\\times H;\\bR)$ by\n\\[F(x,y):=\\nnrm{G''(x)-G''(y)}{\\ensuremath{\\mathscr L}(H)},\\quad x,\\,y\\in H,\\]\nand fix $(t,x)\\in[0,T]\\times H$, $((t_k,x_k))_{k\\in\\bN}\\subset[0,T]\\times H$ with $(t_k,x_k)\\to (t,x)$ as $k\\to\\infty$. Note that $Z(T,t_k,x_k)\\to Z(T,t,x)$ in $L^2(\\Omega;H)$ by It\\^{o}'s isometry. As a consequence, $(Z(T,t,x),Z(T,t_k,x_k))\\to (Z(T,t,x),Z(T,t,x))$ in distribution (on $H\\times H$) and we obtain\n\\begin{align*}\n\\nnrm{u_{xx}(t,x)-u_{xx}(t_k,x_k)}{\\ensuremath{\\mathscr L}(H)}\n&\\leq \\bE\\, F\\big(Z(T,t,x),Z(T,t_k,x_k)\\big)\\\\\n&\\xrightarrow{k\\to\\infty}\\bE\\, F\\big(Z(T,t,x),Z(T,t,x)\\big)=0,\n\\end{align*}\nyielding the continuity of $u_{xx}$.\n\nBy Taylor's formula and Lemma~\\ref{lem:integrandIsomorphism},\n\\begin{equation*}\n\\begin{aligned}\n\\int_{U_1} \\big|u\\big(t,x+E(T-t)By&\\big)- u(t,x)-\\big\\langle u_x(t,x),E(T-t)By\\big\\rangle_H\\big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\;\\frac12\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_{U_1}\\nnrm{E(T-t)By}{H}^2\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&=\\;\\frac12\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{E(T-t)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2.\n\\end{aligned}\n\\end{equation*}\nUsing Assumption~\\ref{ass:abstractSetting}(v), condition \\eqref{eq:backwardKolm0} follows with $$C_\\epsilon=1\/2\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{\\Phi_\\epsilon}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2.$$\n\nIn order to verify the Kolmogorov equation \\eqref{eq:backwardKolm1}, we first note that for fixed $t\\in[0,T]$ the $H$-valued random variables $\\int_0^tE(s)B\\,\\ensuremath{\\mathrm{d}} L(s)$ and $\\int_{T-t}^{T} E(T-s)B\\,\\ensuremath{\\mathrm{d}} L(s)$ have the same distribution, so that\n\\begin{equation}\\label{eq:backwardKolm2}\nv(t,x):=\\bE\\,G\\big(x+\\int_0^{t}E(s)B\\,\\ensuremath{\\mathrm{d}} L(s)\\big)=u(T-t,x),\\quad (t,x)\\in [0,T]\\times H.\n\\end{equation}\nNext, we fix $x\\in H$ and apply It\\^{o}'s formula \\cite[Theorem~3.6]{MRT13} to the function $y\\mapsto G(x+y)$ and the martingale $M=(M(t))_{t\\in[0,T]}:=(\\int_0^t E(s)B\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}\\in\\cM^2_T(H)$. Note that $M$ fits into the setting of \\cite{MRT13} since it has the representation\n\\[M(t)=\\int_0^t\\int_{U_1} E(s)By\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y),\\quad t\\in[0,T],\\]\nwhere $q$ is the compensated Poisson random measure on $[0,\\infty)\\times U_1$ associated to $L$; see the appendix for details.\nWe obtain\n\\begin{equation}\\label{eq:backwardKolm3}\n\\begin{aligned}\nG(x+M(t))= G(x)&+\\int_0^t\\int_{U_1}\\big\\{G\\big(x+M(s-)+E(s)By\\big)-G\\big(x+M(s-)\\big)\\big\\}\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y)\\\\\n&+\\int_0^t\\int_{U_1}\\big\\{G\\big(x+M(s)+E(s)By\\big)-G\\big(x+M(s)\\big)\\\\\n&\\hspace{4.4cm}-\\big\\langle G'\\big(x+M(s)\\big),E(s)By\\big\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s,\n\\end{aligned}\n\\end{equation}\nwhere the integrand appearing in the integral w.r.t.\\ $q$ belongs to $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;\\bR)$ as a consequence of Taylor's formula, the boundedness of $G'$, Lemma~\\ref{lem:integrandIsomorphism} and \\eqref{eq:assEB1}. Similarly, the second integral in \\eqref{eq:backwardKolm3} exists for all $\\omega\\in\\Omega$ and belongs to $L^1(\\Omega;\\bR)$\nsince\n\\begin{equation*}\\label{eq:backwardKolm4}\n\\begin{aligned}\n\\int_0^t\\int_{U_1}\\big|G\\big(x+M(s)+E(s)By\\big)-G\\big(&x+M(s)\\big)-\\big\\langle G'\\big(x+M(s)\\big),E(s)By\\big\\rangle_H\\big|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s\\\\\n&\\leq\\;\\frac12\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^t\\nnrm{E(s)B}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\ensuremath{\\mathrm{d}} s.\n\\end{aligned}\n\\end{equation*}\nTaking expectations in \\eqref{eq:backwardKolm3} and using the martingale property of the integral w.r.t.\\ $q$ yields\n\\begin{equation}\\label{eq:backwardKolm5}\nv(t,x)=G(x)+\\int_0^t\\int_{U_1}\\big\\{v(s,x+E(s)By)-v(s,x)-\\langle v_x(s,x),E(s)By\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s.\n\\end{equation}\nBy the fundamental theorem of calculus, \\eqref{eq:backwardKolm1} follows from \\eqref{eq:backwardKolm2}, \\eqref{eq:backwardKolm5} if the mapping\n\\begin{equation}\\label{eq:backwardKolm7}\n(0,T]\\ni s\\to \\int_{U_1}\\big\\{v(s,x+E(s)By)-v(s,x)-\\langle v_x(s,x),E(s)By\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\in\\bR.\n\\end{equation}\nis continuous.\n\nNote that we cannot apply directly the continuity theorem for parameter-dependent integrals to show the continuity of the mapping \\eqref{eq:backwardKolm7}. The reason is that the term $E(s)By$ in the integral in \\eqref{eq:backwardKolm7} is defined only in an $L^2([0,T]\\times U_1,\\lambda\\otimes\\nu;H)$-sense, cf.~Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix}, so that we have no information about the continuity of $(0,T]\\in s\\mapsto E(s)By\\in H$ for fixed $y\\in U_1$. Therefore, we use an approximation argument: For $s\\in(0,T]$, $x\\in H$, $y\\in U_1$ and $k\\in\\bN$ set\n\\begin{align*}\nf(s,x,y)&:=v(s,x+E(s)By)-v(s,x)-\\langle v_x(s,x),E(s)By\\rangle_H,\\\\\nf_k(s,x,y)&:=f(s,x,\\Pi_ky),\n\\end{align*}\nwhere $\\Pi_k$ is the orthogonal projection of $U_1$ onto $\\text{span}\\{f_1,\\ldots,f_k\\}$, $(f_k)_{k\\in\\bN}\\subset U_0$ being an orthonormal basis of $U_1$ as in Lemma~\\ref{lem:integrandIsomorphism}. For fixed $x\\in H$,  $f(s,x,y)$ is defined in an $L^2([0,T]\\times U_1,\\lambda\\otimes\\nu;\\bR)$-sense whereas $f_k(s,x,y)$ is defined pointwise. The continuity theorem for parameter-dependent integrals and the strong continuity of $(E(t))_{t\\geq 0}$ yield the continuity of in $\\int_{U_1}f_k(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ in $(s,x)\\in[0,T]\\times H$. Moreover, we have $f_k(s,x,\\cdot)\\stackrel{k\\to\\infty}{\\longrightarrow} f(s,x,\\cdot)$ in $L^1(U_1,\\nu;\\bR)$, uniformly in $(s,x)\\in[\\epsilon,T]\\times H$ for all $\\epsilon>0$. Indeed, setting $\\Pi^ky:=y-\\Pi_ky$ and using Taylor's theorem, Lemma~\\ref{lem:integrandIsomorphism} and Assumption~\\ref{ass:abstractSetting}(v), we obtain\n\\begin{align*}\n&\\int_{U_1}|f(s,x,y)-f_k(s,x,y)|\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\int_{U_1}\\int_0^1\\big|\\big\\langle v_{xx}\\big(s,x+E(s)B(\\Pi_ky+\\theta\\Pi^ky)\\big) E(s)B\\Pi^ky,E(s)B\\Pi^ky\\big\\rangle_H\\big|(1-\\theta)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\quad+\\int_{U_1}\\int_0^1\\big|\\big\\langle v_{xx}\\big(s,x+\\theta E(s)B\\Pi_ky\\big)E(s)B\\Pi_ky,E(s)B\\Pi^ky\\big\\rangle_H\\big|\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\big(\\nnrm{E(s)B\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2+\\nnrm{E(s)B\\Pi_k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\nnrm{E(s)B\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\big)\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}\\big(\\nnrm{\\Phi_\\epsilon\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2+\\nnrm{\\Phi_\\epsilon\\Pi_k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\nnrm{\\Phi_\\epsilon\\Pi^k}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\big)\n\\end{align*}\nfor all $s\\in[\\epsilon,T]$ and some $\\Phi_\\epsilon\\in\\ensuremath{\\mathscr L_2}(U_0,H)$. The expression in the last line tends to zero as $k\\to\\infty$.\nAs a consequence, $\\int_{U_1}f_k(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)\\xrightarrow{k\\to\\infty}\\int_{U_1}f(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ uniformly in $(s,x)\\in[\\epsilon,T]\\times H$. Thus, the continuity of $\\int_{U_1}f_k(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ in $(s,x)\\in[0,T]\\times H$ implies the continuity of $\\int_{U_1}f(s,x,y)\\nu(\\ensuremath{\\mathrm{d}} y)$ in $(s,x)\\in(0,T]\\times H$. In particular, we obtain the continuity of the mapping \\eqref{eq:backwardKolm7} as well as the continuity of $u_t$ on $[0,T)\\times H$.\n\\end{proof}\n\n\nFor $G\\in C_{\\operatorname b}^2(H,\\bR)$, the regularity assertions in Propostition~\\ref{prop:backwardKolm} allow us to apply It\\^{o}'s formula \\cite[Theorem~3.6]{MRT13} to the function $(t,x)\\mapsto u(t,x)$ and the $H$-valued martingale $\\tilde Y=(\\tilde Y(t))_{t\\in[0,T]}$ defined in \\eqref{eq:defYtilde}. Note that $\\tilde Y$ fits into the setting of \\cite{MRT13} since it has the representation\n\\begin{equation}\\label{eq:proofErrRep1.5}\n\\tilde Y(t)=\\tilde E(T)X_0+\\int_0^t\\int_{U_1}\\tilde E(T-s) B y\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y),\\quad t\\in[0,T],\n\\end{equation}\nwhere again $q$ is the compensated Poisson random measure on $[0,\\infty)\\times U_1$ associated with $L$ as described in the appendix. Equality \\eqref{eq:proofErrRep1.5} is a consequence of \\eqref{eq:assEBtilde1}, Lemma~\\ref{lem:integrandIsomorphism}, Remark~\\ref{not:Phix} and\nLemma~\\ref{lem:comparisonIntegrals}. For $T'\\in(0,T)$ we obtain\n\\begin{equation}\\label{eq:proofErrRep2}\n\\begin{aligned}\nu(T',\\tilde Y(T'))\\,&=\\,u(0,\\tilde Y(0))+\\int_0^{T'}u_t(t,\\tilde Y(t))\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\quad+\\int_0^{T'}\\int_{U_1}\\big\\{u\\big(t,\\tilde Y(t-)+\\tilde E(T-t) By\\big)-u(t,\\tilde Y(t-))\\big\\}\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} y)\\\\\n&\\quad+\\int_0^{T'}\\int_{U_1}\\big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u(t,\\tilde Y(t))\\\\\n&\\quad-\\big\\langle u_x(t,\\tilde Y(t)),\\tilde E(T-t) By\\big\\rangle_H\\big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} s.\n\\end{aligned}\n\\end{equation}\nUsing the boundedness of $u$, $u_x$ and $u_{xx}$, \\eqref{eq:backwardKolm1}, \\eqref{eq:assEB1} and applying similar arguments as in the proof of Proposition~\\ref{prop:backwardKolm}, one sees that all terms in \\eqref{eq:proofErrRep2} are well-defined and integrable w.r.t.\\ $\\bP$. Thus, we can take expectations and use the martingale property of the integral w.r.t.\\ $q$ and the backward Kolmogorov equation \\eqref{eq:backwardKolm1} to obtain\n\\begin{equation}\\label{eq:proofErrRep3}\n\\begin{aligned}\n\\bE\\big(&u(T',\\tilde Y(T'))-u(0,\\tilde Y(0))\\big)=\\\\\n&\\bE\\int_0^{T'}\\int_{U_1}\\Big\\{u\\big(t,\\tilde Y(t)+\\tilde E(T-t) By\\big)-u\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\qquad\\qquad\\qquad\\qquad-\\big\\langle u_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big\\}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\n\\end{aligned}\n\\end{equation}\nfor all $T'\\in(0,T)$. Taking the limit $T'\\to T$ on both sides of \\eqref{eq:proofErrRep3}, we can replace $T'$ by $T$. Here we used the stochastic continuity of $\\tilde Y$ and the continuity of $u$ for the limit on the left hand side. For the limit on the right hand side we used \\eqref{eq:errorRep1}, which is again a consequence of Taylor's formula, the boundedness of $G''$, \\eqref{eq:assEB1}, \\eqref{eq:assEBtilde1} and Lemma~\\ref{lem:integrandIsomorphism}, using similar arguments as in the proof of Proposition~\\ref{prop:backwardKolm}. The combination of \\eqref{eq:proofErrRep1} and \\eqref{eq:proofErrRep3} yields the error representation formula \\eqref{eq:errorRep2} for the case $G\\in C_{\\operatorname b}^2(H,\\bR)$.\n\nFinally, we consider the general case of a test function $G\\in C^2(H,\\bR)$ such that $\\sup_{x\\in H}\\nnrm{G''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$.\nFor $\\epsilon>0$, define $G_\\epsilon\\in C_{\\operatorname b}^2(H,\\bR)$ by\n\\begin{equation*}\nG_\\epsilon(x):=e^{-\\epsilon\\nnrm{x}H^2} G(x),\\quad x\\in H.\n\\end{equation*}\nThe Fr\\'{e}chet derivatives $G_\\epsilon'(x)\\in H$ and $G_\\epsilon''(x)\\in\\ensuremath{\\mathscr L}(H)$ are given by\n\\begin{align*}\nG_\\epsilon'(x)&=e^{-\\epsilon\\nnrm{x}H^2}\\big(G'(x)-2\\epsilon G(x)x\\big),\\\\\nG_{\\epsilon}''(x)&=e^{-\\epsilon\\nnrm{x}H^2}\\big[G''(x)+2\\epsilon\\big(\\langle G'(x),\\arg\\rangle_H x-\\langle x,\\arg\\rangle_H G'(x)\\big)-4\\epsilon^2 G(x)\\langle x,\\arg\\rangle_Hx\\big],\n\\end{align*}\nso that the boundedness of $G_\\epsilon:H\\to\\bR$, $G_\\epsilon':H\\to H$ and $G_\\epsilon'':H\\to\\ensuremath{\\mathscr L}(H)$ follows from \\eqref{eq:growthGG'}. Moreover, note that $G_\\epsilon(x)\\xrightarrow{\\epsilon\\to0}G(x)$ in $\\bR$ and $G_\\epsilon'(x)\\xrightarrow{\\epsilon\\to0}G'(x)$ in $H$ for all $x\\in H$, as well as\n$\n\\sup_{\\epsilon\\in(0,1]}\\sup_{x\\in H}\\nnrm{G_\\epsilon''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty.\n$\nThe latter is a consequence of \\eqref{eq:growthGG'} and the standard estimate $\\sup_{s>0}s^ne^{-s}<n!$, $n\\in\\bN$.  We set\n\\begin{equation}\\label{eq:defuxtEps}\nu_\\epsilon(t,x):=\\bE\\,G_\\epsilon(Z(T,t,x)),\\quad (t,x)\\in [0,T]\\times H.\n\\end{equation}\nBy what has been shown above, the assertion of Theorem~\\ref{thm:errorRep} holds with $u$ replaced by $u_\\epsilon$. By a dominated convergence argument using \\eqref{eq:growthGG'}, we obtain that $u_\\epsilon(t,x)\\xrightarrow{\\epsilon\\to0} u(t,x)$ in $\\bR$ and $(u_\\epsilon)_x(t,x)\\xrightarrow{\\epsilon\\to0} u_x(t,x)$ in $H$ for all $(t,x)\\in [0,T]\\times H$. For all $t\\in[0,T]$ and $y\\in U_1$, we have\n\\begin{equation}\\label{eq:proofErrRep4}\n\\begin{aligned}\n\\Big|u_\\epsilon\\big(t,&\\tilde Y(t)+\\tilde E(T-t) By\\big)-u_\\epsilon\\big(t,\\tilde Y(t)+E(T-t)By\\big)\\\\\n&\\qquad-\\big\\langle (u_\\epsilon)_x(t,\\tilde Y(t)),\\big(\\tilde E(T-t) B-E(T-t)B\\big)y\\big\\rangle_H\\Big|\\\\\n&\\leq \\sup_{\\epsilon\\in(0,1]}\\sup_{x\\in H}\\nnrm{G_\\epsilon''(x)}{\\ensuremath{\\mathscr L}(H)}\\Big(\\nnrm{\\tilde E(T-t) By}H^2+\\nnrm{E(T-t)B}H^2\\big)\n\\end{aligned}\n\\end{equation}\ndue to Taylor's theorem and the fact that $(u_\\epsilon)_{xx}(t,x)=\\bE G_\\epsilon''(Z(T,t,x))$. Note that the term right hand side of \\eqref{eq:proofErrRep4} is integrable w.r.t. $\\ensuremath{\\mathrm{d}} t\\,\\nu(\\ensuremath{\\mathrm{d}} y)$ as a consequence of \\eqref{eq:assEB1}, \\eqref{eq:assEBtilde1} and Lemma~\\ref{lem:integrandIsomorphism}. Thus, considering \\eqref{eq:errorRep1} and \\eqref{eq:errorRep2} with $u$ replaced by $u_\\epsilon$, we can use dominated convergence as $\\epsilon\\to0$  to finish the proof.\n\n\n\\section{Applications to parabolic equations}\n\\label{sec:Heat}\n\n\\subsection{The stochastic heat equation}\\label{subsec:she} Here we give a detailed error analysis of a space-time discretization of the linear stochastic heat equation with additive L\\'{e}vy noise.\n\nLet $\\ensuremath{\\mathcal{O}}\\subset\\bR^d$ be a bounded convex domain. \nLet $\\Lambda:=-\\Delta=-\\sum_{j=1}^d\\partial^2\/\\partial\\xi_j^2$ be the Laplace operator on $L^2(\\ensuremath{\\mathcal{O}})$ with zero-Dirichlet boundary condition, i.e., with domain $D(\\Lambda):=\\{v\\in H^1_0{(\\ensuremath{\\mathcal{O}})}:~\\Lambda u\\in L^2(\\ensuremath{\\mathcal{O}})\\}$, where $\\Lambda u$ is understood in the distributional sense, see \\cite[Example 3.4.7]{ABHN11}. As usual, $H^n(\\ensuremath{\\mathcal{O}})$ denotes the $L^2$-Sobolev space of order $n\\in\\bN_0$ on $\\ensuremath{\\mathcal{O}}$  and $H^1_0(\\ensuremath{\\mathcal{O}})$ is the $H^1(\\ensuremath{\\mathcal{O}})$-closure of the space $C_c^\\infty(\\ensuremath{\\mathcal{O}})$ of compactly supported test functions.\nThen, setting\n\\[H:=U:=L^2(\\ensuremath{\\mathcal{O}}),\\quad (A,D(A)):=(\\Lambda,D(\\Lambda)),\\quad B:=\\id{L^2(\\ensuremath{\\mathcal{O}})},\\]\nthe abstract equation \\eqref{eq:mainEq} becomes the stochastic heat equation \\eqref{eq:SHE}. It is not difficult to see that the condition $\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty$ implies \\eqref{eq:assEB1}, where\n\\begin{equation}\\label{eq:E(t)SHE}\n(E(t))_{t\\geq0}:=(e^{-t\\Lambda})_{t\\geq0}\\subset\\ensuremath{\\mathscr L}(H)\n\\end{equation}\nis the semigroup generated by $-A=-\\Lambda$. Hence, there exists a unique weak solution $X=(X(t))_{t\\geq0}$ to Eq.~\\eqref{eq:SHE}, given by the variation-of-constants formula \\eqref{eq:X}. Furthermore, for some $C>0$ independent of $T$,\n\\begin{equation}\\label{eq:hstab}\n\\nnrm{X(T)}{L^2(\\Omega,H)}\\le C (\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}( H)}+\\nnrm{X_0}{L^2(\\Omega,H)}).\n\\end{equation}\nIn the sequel, we use the smoothness spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, defined by\n\\begin{align*}\n\\dot H^\\alpha&:=D(\\Lambda^{\\alpha\/2})\\\\\n&:=\\Big\\{v=\\sum_{k=1}^\\infty v_k\\varphi_k:(v_k)_{k\\in\\bN}\\subset\\bR,\\;|v|_{\\alpha}:=\\nnrm{\\Lambda^{\\alpha\/2}v}{L^2(\\ensuremath{\\mathcal{O}})}=\\Big(\\sum_{k=1}^\\infty \\lambda_k^\\alpha v_k^2\\Big)^{1\/2}<\\infty\\Big\\},\n\\end{align*}\nwhere $(\\varphi_k)_{k\\in\\bN}\\subset D(\\Lambda)$ is an orthonormal basis of $L^2(\\ensuremath{\\mathcal{O}})$ consisting of eigenfunctions of $\\Lambda $ and $(\\lambda_k)_{k\\in\\bN}\\subset(0,\\infty)$ is the corresponding sequence of eigenvalues; compare \\cite[Chapters 3 and 19]{Tho06}. They are Hilbert spaces and one has the identities $\\dot H^0=H=L^2(\\ensuremath{\\mathcal{O}})$, $\\dot H^1=H^1_0(\\ensuremath{\\mathcal{O}})$ and $\\dot H^2=D(\\Lambda)=H^2(\\ensuremath{\\mathcal{O}})\\cap H^1_0(\\ensuremath{\\mathcal{O}})$, where the natural norms of the respective spaces are equivalent. The latter equality is a consequence of the elliptic regularity estimate\n\\begin{equation}\\label{eq:ellreg}\n\\|v\\|_{H^2}\\le C\\|v\\|_{\\dot H^2},\\quad v\\in \\dot{H}^2,\n\\end{equation}\nfor bounded convex domains, see \\cite[Corollary 1]{Fromm93}. Without the convexity assumption one can still define the $\\dot H^\\alpha$ spaces as above but one does not obtain a characterization of $D(\\Lambda)=\\dot{H}^2$ in terms of classical Sobolev spaces.\nFor negative $\\alpha$, the elements of $\\dot H^\\alpha$ are formal sums and we identify them with elements of $L^2(\\ensuremath{\\mathcal{O}})$ if $\\sum_{k=1}^\\infty v_k^2<\\infty$, so that $\\dot H^\\alpha$ is the closure of $L^2(\\ensuremath{\\mathcal{O}})$ w.r.t.\\ the $|\\cdot|_\\alpha$-norm.\n\\begin{remark}\\label{rem:interpolation}\nThe spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, can be obtained by both real and complex interpolation: For $\\alpha=(1-\\theta)\\alpha_0+\\theta\\alpha_2$, $\\theta\\in(0,1)$, one has $\\dot H^\\alpha=(\\dot H^{\\alpha_0},\\dot H^{\\alpha_1})_{\\theta,2}=[\\dot H^{\\alpha_0},\\dot H^{\\alpha_1}]_\\theta$ with equivalent norms, where $(\\cdot,\\cdot)_{\\theta,2}$ and $[\\cdot,\\cdot]_\\theta$ denotes real interpolation with summability parameter $q=2$ and complex interpolation, respectively. This follows, e.g., from \\cite[Theorem 1.18.5]{Tri78} and the fact that the spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, are isometrically isomorphic to weighted $\\ell^2$-spaces. We will frequently use the corresponding interpolation inequalities in this and the next section.\n\\end{remark}\n\nFor the spatial discretization of Eq.~\\eqref{eq:SHE}, we consider a family of finite-dimensional spaces $(S_h)_{h>0}\\subset H^1_0(\\ensuremath{\\mathcal{O}})$.\n Unless otherwise stated, we endow the spaces $S_h$ with the inner product $\\langle\\cdot,\\cdot\\rangle_H$ and the norm $\\nnrm{\\cdot}H$.  By $P_h:H\\to S_h$ and $\\Pi_h:\\dot H^1\\to S_h$ we denote the orthogonal projections with respect to the inner products in $H$ and $\\dot H^1$, respectively. The discrete Laplacian $\\Lambda_h:S_h\\to S_h$ is defined by\n\\begin{equation}\\label{eq:discreteLaplace}\n\\langle\\Lambda_h v,w\\rangle_{L^2(\\ensuremath{\\mathcal{O}})}=\\langle\\nabla v,\\nabla w\\rangle_{L^2(\\ensuremath{\\mathcal{O}};\\bR^2)},\\quad v,w\\in S_h.\n\\end{equation}\nOur assumption on the spatial approximation is formulated via the following estimate on the Ritz projection $\\Pi_h$,\n\\begin{equation}\\label{eq:basicFEMestimate1}\n\\nnrm{\\Pi_hv-v}{L^2(\\ensuremath{\\mathcal{O}})}\\leq C h^\\beta|v|_\\beta,\\quad v\\in\\dot H^\\beta,\\;1\\leq\\beta\\leq2.\n\\end{equation}\nThis holds for example, if $S_h$ is consisting of piecewise linear functions with respect to a family of triangulations of $\\ensuremath{\\mathcal{O}}$. The parameter $h$ corresponds to the maximal mesh size of the triangulation, see, e.g., \\cite[Lemma~1.1]{Tho06} or \\cite[Section~5.4]{LarTho03}.\n\nThe time discretization of Eq.~\\eqref{eq:SHE} on a finite interval $[0,T]$ is done via the implicit Euler scheme with time step $\\ensuremath{\\Delta t}=T\/N$, $N\\in\\bN$, and grid points $t_n=n\\ensuremath{\\Delta t}$, $n=0,\\ldots N$. For $h>0$ and $N\\in\\bN$, the discretization $(X^n_{h,\\ensuremath{\\Delta t}})_{n=1,\\ldots,N}$ of $(X(t))_{t\\in[0,T]}$ in space and time is given as the solution to\n\\begin{equation}\\label{eq:schemeSHE}\nX^n_{h,\\ensuremath{\\Delta t}}-X^{n-1}_{h,\\ensuremath{\\Delta t}}+\\ensuremath{\\Delta t}\\Lambda_hX^n_{h,\\ensuremath{\\Delta t}}=P_h(L(t_n)-L(t_{n-1})),\\quad n=1,\\ldots,N;\\quad X^0_{h,\\ensuremath{\\Delta t}}=P_hX_0.\n\\end{equation}\n\n\\begin{remark}[strong error] \\label{rem:strongErrorSHE}\nIf the covariance operator $Q\\in\\ensuremath{\\mathscr L}(H)$ of $L$ is such that\n\\begin{equation}\\label{eq:SHEassQ}\n\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty\n\\end{equation}\nfor some $\\beta\\geq0$, then the solution $X(t)$ takes values in $\\dot H^\\beta$ for all $t>0$. For the Gaussian case, i.e., the case where $L$ in \\eqref{eq:SHE} is a $Q$-Wiener process, it has been shown in \\cite[Theorem~1.2]{Yan05} that, if \\eqref{eq:SHEassQ} holds and $X_0\\in L^2(\\Omega,\\cF_0,\\bP;\\dot H^\\beta)$ for some $\\beta\\in(0,1]$, then the scheme \\eqref{eq:schemeSHE} has strong convergence of order $\\beta$ in space and $\\beta\/2$ in time:\n\\[\\nnrm{X^n_{h,\\ensuremath{\\Delta t}}-X(t_n)}{L^2(\\Omega;H)}\\leq C(h^\\beta+(\\ensuremath{\\Delta t})^{\\frac\\beta2}),\\quad n=0,\\ldots,N.\\]\nUnlike weak error estimates, strong $L^2$-error estimates are the same in the Gaussian case and in our setting, since the only stochastic tool that is needed is It\\^{o}'s isometry~\\eqref{eq:ItoIsom} which looks the same\nif the driving noise is a L\\'{e}vy process which is an $L^2$-martingale.\nThus the strong error result in \\cite[Theorem~1.2]{Yan05} carries over one-to-one to our setting.\n\\end{remark}\n\n\\begin{remark}\nThe $S_h$-valued random variables $P_h(L(t_n)-L(t_{n-1}))$ in \\eqref{eq:schemeSHE} can be defined in two ways. On the one hand, we may set\n\\[P_h(L(t_n)-L(t_{n-1})):=L^2(\\Omega;S_h)\\text{-}\\lim_{K\\to\\infty}\\sum_{k=1}^K(L_k(t_n)-L_{k}(t_{n-1}))P_he_k,\\]\nwith an orthonormal basis $(e_k)_{k\\in\\bN}$ of $U_0$ and real-valued uncorrelated L\\'{e}vy processes $L_k=(L_k(t))_{t\\geq0}$, $k\\in\\bN$, as in Remark~\\ref{rem:approxL}. The limit exists since, by the finite-dimensionality of $S_h$, one has $P_h\\in\\ensuremath{\\mathscr L_2}(H,S_h)=\\ensuremath{\\mathscr L_2}(U,S_h)\\subset\\ensuremath{\\mathscr L_2}(U_0,S_h)$.\nOn the other hand, we can extend the orthogonal projection $P_h:H\\to S_h$ to a generalized $L^2$-projection $P_h:\\dot H^{-1}\\to S_h$ defined by\n\\[\\langle P_hv,w\\rangle_{H}=\\langle v,w\\rangle_{\\dot H^{-1}\\times\\dot H^1},\\quad v\\in \\dot H^{-1},\\;w\\in S_h.\\] Then, the assumption $\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty$ implies that we can take $U_1:=D(\\Lambda^{-1\/2})=\\dot H^{-1}$ as the state space of $L$, so that the expression $P_h(L(t_n)-L(t_{n-1}))$ makes sense $\\omega$-wise. Obviously, both definitions are compatible.\nIn practice, one has to find a suitable way to sample (an approximation of) the discretized noise increment $P_h(L(t_n)-L(t_{n-1}))$. We do not treat this problem in the present paper but refer to \\cite{BarLan12a, DunHauPro12} and \\cite[Remark~4]{LinSch13} for related considerations.\n\\end{remark}\n\n  With $R(\\lambda):=1\/(1+\\lambda)$ and $E_{h,\\ensuremath{\\Delta t}}:=R(\\ensuremath{\\Delta t}\\Lambda_h):=(I+\\ensuremath{\\Delta t}\\Lambda_h)^{-1}$ as well as $E_{h,\\ensuremath{\\Delta t}}^n:=R^n(\\ensuremath{\\Delta t}\\Lambda_h):=((I+\\ensuremath{\\Delta t}\\Lambda_h)^{-1})^n$, the scheme \\eqref{eq:schemeSHE} can be rewritten as\n\\begin{equation}\\label{eq:schemeSHE2}\nX^n_{h,\\ensuremath{\\Delta t}}=E_{h,\\ensuremath{\\Delta t}}^nP_hX_0+\\sum_{j=1}^n E^{n-j+1}_{h,\\ensuremath{\\Delta t}}P_h(L(t_j)-L(t_{j-1})),\\quad n=0,\\ldots,N.\n\\end{equation}\nFor $t\\in[0,T]$, let $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)\\in\\ensuremath{\\mathscr L}(H)$ be defined by\n\\begin{equation}\\label{eq:EtildeSHE}\n\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t):=\\mathds 1_{\\{0\\}}(t)P_h+\\sum_{n=1}^N\\mathds 1_{(t_{n-1},t_n]}(t)E_{h,\\ensuremath{\\Delta t}}^nP_h\n\\end{equation}\nand set\n\\begin{equation}\\label{eq:XtildeSHE}\n\\tilde X(t)=\\tilde X_{h,\\ensuremath{\\Delta t}}(t):=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)X_0+\\int_0^t\\tilde E_{h,\\ensuremath{\\Delta t}}(t-s)\\,\\ensuremath{\\mathrm{d}} L(s).\n\\end{equation}\nThen $X^n_{h,\\ensuremath{\\Delta t}}=\\tilde X_{h,\\ensuremath{\\Delta t}}(t_n)$ $\\bP$-almost surely. This follows from the construction of the stochastic integral, using an approximation argument and It\\^{o}'s isometry \\eqref{eq:ItoIsom}.\n\nThe following deterministic estimates will be used in the proof of our weak error result stated in Theorem~\\ref{thm:SHE} below.\n\\begin{lemma}\nThe operators $E(t)$ and $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)$ defined in \\eqref{eq:E(t)SHE} and \\eqref{eq:EtildeSHE} satisfy the error estimates\n\\begin{align}\n\\nnrm{\\tilde E(s)-E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C(h^{2}+\\ensuremath{\\Delta t})s^{-1},\\label{eq:detEstSHE1}\\\\\n\\nnrm{\\Lambda^\\alpha E(s)}{\\ensuremath{\\mathscr L}(H)}+\\nnrm{\\Lambda^\\alpha \\tilde E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C s^{-\\alpha},\\quad 0\\leq\\alpha\\leq1\/2,\\label{eq:detEstSHE2}\n\\end{align}\n$s\\in(0,T]$, where $C>0$ does not depend on $h$, $\\ensuremath{\\Delta t}$ and $s$.\n\\end{lemma}\n\n\\begin{proof}\nEstimate \\eqref{eq:detEstSHE1} follows from\n\\begin{equation}\\label{eq:hest}\n\\nnrm{E_{h,\\ensuremath{\\Delta t}}^nP_h-E(t_n)}{\\ensuremath{\\mathscr L}(H)}\\leq C(h^2+\\ensuremath{\\Delta t})t_n^{-1},\n\\end{equation}\nsee, for example, \\cite[Theorem~7.7]{Tho06}. We note here that while the latter result is proved under the assumption that $\\ensuremath{\\mathcal{O}}$ has smooth boundary, the proof relies on the availability of \\eqref{eq:basicFEMestimate1}, which is our basic assumption, and hence \\eqref{eq:hest} is valid in our setting.\nFor $s\\in(t_{n-1},t_n]$ we have\n\\begin{align*}\n\\nnrm{(E(t_n)-E(s))v}H&=\\nnrm{\\Lambda E(s)(E(t_n-s)-\\id{H})\\Lambda^{-1}v}H\\\\\n&\\leq\\nnrm{\\Lambda E(s)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{(E(t_n-s)-\\id{H})\\Lambda^{-1}v}H\\\\\n&\\leq Cs^{-1}\\ensuremath{\\Delta t}\\nnrm{v}H,\n\\end{align*}\nwhere we used Theorem~6.13(c),(d) on analytic semigroups in \\cite[Chapter 2]{Paz83}.\nEstimate~\\eqref{eq:detEstSHE2} is due to Theorem~6.13(c) in \\cite[Chapter 2]{Paz83}, Lemma~7.3 in  \\cite{Tho06}, interpolation, and the fact that $\\nrm{A^\\alpha v_h}\\leq\\nrm{A^\\alpha_hv_h}$ for $v_h\\in S_h$, $0\\leq\\alpha\\leq 1\/2$. The latter follows from the basic identity $\\nrm{A^{1\/2}v_h}=\\nrm{A_h^{1\/2}v_h}$ and interpolation.\n\\end{proof}\n\n\nHere is our result for the weak error of the discretization of the stochastic heat equation.\n\\begin{theorem}\\label{thm:SHE}\nAssume that $X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$ and $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}<\\infty$ for some $\\beta\\in(0,1]$. Let $(X(t))_{t\\geq0}$ be the weak solution \\eqref{eq:X} to Eq.~\\eqref{eq:mainEq} and let $(X_{h,\\ensuremath{\\Delta t}}^n)_{n=0,\\ldots,N}$ be defined by the scheme \\eqref{eq:schemeSHE}. Given $g\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$, there exists a constant $C=C(g,T)>0$\nthat does not depend on $h$ and $\\ensuremath{\\Delta t}$, such that\n\\[\\big|\\bE\\big(g(X^N_{h,\\ensuremath{\\Delta t}})-g(X(T))\\big)\\big|\\leq C(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta)|\\log(h^2+\\ensuremath{\\Delta t})|\\]\nfor $h^{2}+ \\ensuremath{\\Delta t}\\leq 1\/e$.\n\\end{theorem}\n\n\\begin{proof}\nWe are in the setting of Section~\\ref{sec:Setting and preliminaries} with $H=U=L^2(\\ensuremath{\\mathcal{O}})$, $B=\\id H$, and $(E(t))_{t\\geq0}$, $(\\tilde E(t))_{t\\in[0,T]}=(\\tilde E_{h,\\ensuremath{\\Delta t}}(t))_{t\\in[0,T]}$, $(\\tilde X(t))_{t\\in[0,T]}=(\\tilde X_{h,\\ensuremath{\\Delta t}}(t))_{t\\in[0,T]}$ being given by \\eqref{eq:E(t)SHE}, \\eqref{eq:EtildeSHE}, \\eqref{eq:XtildeSHE} respectively. In particular, Assumption~\\ref{ass:abstractSetting} is fulfilled. Since $X_{h,\\ensuremath{\\Delta t}}^N=\\tilde X(T)$, we can use Corollary~\\ref{cor:errorRep} with $G:=g$ to estimate the weak error.\nLet $F(t):=\\tilde E(t)-E(t)$ be the deterministic error operator.\n\nWe begin with the first term on the right hand side of \\eqref{eq:errorRep4} in Corollary~\\ref{cor:errorRep}.\nThe stability estimate \\eqref{eq:hstab} and the deterministic estimate \\eqref{eq:detEstSHE1} yield, for $\\max(h^{2},\\ensuremath{\\Delta t})\\leq 1$,\n\\begin{equation}\\label{eq:proofSHE1}\n\\begin{aligned}\n&\\big|\\bE\\big\\{u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\big|\\\\\n&= \\big|\\bE\\big\\{u(0,\\tilde Y(0))-u(0,Y(0))\\big\\}\\big|\\\\\n&= \\Big|\\bE\\int_0^1\\big\\langle u_x\\big(0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big),\\tilde Y(0)-Y(0)\\big\\rangle_{ H}\\ensuremath{\\mathrm{d}} \\theta\\Big|\\\\\n&= \\Big|\\bE\\int_0^1\\big\\langle\\bE\\big(g'(Z(T,0,x))\\big)\\big|_{x=Y(0)+\\theta(\\tilde Y(0)-Y(0))},\\tilde Y(0)-Y(0)\\big\\rangle_{H}\\ensuremath{\\mathrm{d}}\\theta\\Big|\\\\\n&\\leq \\int_0^1\\big\\|g'\\big(Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big)\\big\\|_{L^2(\\Omega, H)}\\ensuremath{\\mathrm{d}}\\theta\\, \\|(\\tilde E(T)-E(T))X_0\\|_{L^2(\\Omega, H)}\\\\\n&\\leq C\\big(1+\\int_0^1\\big\\|Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big\\|_{L^2(\\Omega, H)}\\ensuremath{\\mathrm{d}}\\theta\\big)\\,(h^2+\\ensuremath{\\Delta t})\\,T^{-1}\\,\\nnrm{X_0}{L^2(\\Omega; H)}\\\\\n&\\leq C\\big(1+\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}+\\nnrm{X_0}{L^2(\\Omega; H)}\\big)\\,\\nnrm{X_0}{L^2(\\Omega; H)}\\,T^{-1}\\,(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta).\n\\end{aligned}\n\\end{equation}\n\nNext, consider the second term on the right hand side of \\eqref{eq:errorRep4}. We estimate the integrals of the functions $\\Psi_1$ and $\\Psi_2$ separately. Using Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix}, we obtain\n\\begin{equation}\\label{eq:proofSHE2}\n\\begin{aligned}\n\\big|\\bE\\int_0^T&\\int_{U_1}\\int_0^1\\Psi_1(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\big|\\\\\n&\\leq \\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\int_{U_1}\\nnrm{F(T-t)y}H^2\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\\\\n&=\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\nnrm{F(T-t)}{\\ensuremath{\\mathscr L_2}(U_0,H)}^2\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq C \\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta).\n\\end{aligned}\n\\end{equation}\nThe last step is due to the fact that, by It\\^{o}'s isometry \\eqref{eq:ItoIsom}, the integral in the penultimate line is the square of the strong error $\\nnrm{X^N_{h,\\ensuremath{\\Delta t}}-X(T)}{L^2(\\Omega;H)}$ for zero initial condition $X_0=0$, which can be estimated as in the Gaussian case \\cite[Theorem~1.2]{Yan05}, compare Remark~\\ref{rem:strongErrorSHE}. Further, by the Cauchy-Schwarz inequality, Lemma~\\ref{lem:integrandIsomorphism}, and the fact that $U_0=Q^{1\/2}(U)$,\n\\begin{equation}\\label{eq:proofSHE3}\n\\begin{aligned}\n\\big|\\bE&\\int_0^T\\int_{U_1}\\int_0^1\\Psi_2(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\big|\\\\\n&\\leq \\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\int_{U_1}\\nnrm{E(T-t)y}H\\nnrm{F(T-t)y}H\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\int_0^T\\nnrm{E(T-t)}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\nnrm{F(T-t)}{\\ensuremath{\\mathscr L_2}(U_0,H)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(H)}^2\\int_0^T\\nnrm{E(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\nnrm{F(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\,\\ensuremath{\\mathrm{d}} t\n\\end{aligned}\n\\end{equation}\nBy \\eqref{eq:detEstSHE2} we have\n\\begin{equation}\\label{eq:detEstSHE3}\n\\nnrm{E(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}=\\nnrm{\\Lambda^{\\frac{1-\\beta}2}E(t)}{\\ensuremath{\\mathscr L}(H)}\\leq Ct^{-\\frac{1-\\beta}2}\n\\end{equation}\nand\n\\begin{equation}\\label{eq:detEstSHE4}\n\\nnrm{\\Lambda^\\alpha F(t)}{\\ensuremath{\\mathscr L}(H)}\\leq Ct^{-\\alpha},\\quad 0\\leq\\alpha\\leq1\/2.\n\\end{equation}\nInterpolation between \\eqref{eq:detEstSHE1} and \\eqref{eq:detEstSHE4} with $\\alpha=1\/2$ gives\n\\begin{equation}\\label{eq:detEstSHE5}\n\\nnrm{\\Lambda^{\\frac{1-\\beta}2}F(t)}{\\ensuremath{\\mathscr L}(H)}\\leq C\\nnrm{F(t)}{\\ensuremath{\\mathscr L}(H)}^\\beta\\nnrm{\\Lambda^{\\frac12}F(t)}{\\ensuremath{\\mathscr L}(H)}^{1-\\beta}\\leq C(h^2+\\ensuremath{\\Delta t})^\\beta t^{-\\frac{1+\\beta}2}.\n\\end{equation}\nNote that $\\nnrm{F(t)\\Lambda^\\alpha}{\\ensuremath{\\mathscr L}(H)}=\\nnrm{\\Lambda^\\alpha F(t)}{\\ensuremath{\\mathscr L}(H)}$ due to the self adjointness of $\\tilde E(t)$, $E(t)$ and $\\Lambda^\\alpha$. Altogether, using \\eqref{eq:detEstSHE3}, \\eqref{eq:detEstSHE4} and \\eqref{eq:detEstSHE5}, the integral in the last line of \\eqref{eq:proofSHE3} can be estimated by\n\\begin{equation}\\label{eq:proofSHE4}\n\\begin{aligned}\n\\int_0^T&\\nnrm{E(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\nnrm{F(t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(H)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&=\\Big(\\int_0^{h^2+\\ensuremath{\\Delta t}}+\\int_{h^2+\\ensuremath{\\Delta t}}^T\\Big)\\nnrm{\\Lambda^{\\frac{1-\\beta}2}E(t)}{\\ensuremath{\\mathscr L}(H)}\\nnrm{\\Lambda^{\\frac{1-\\beta}2}F(t)}{\\ensuremath{\\mathscr L}(H)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq C\\int_0^{h^2+\\ensuremath{\\Delta t}}t^{-\\frac{1-\\beta}2}t^{-\\frac{1-\\beta}2}\\,\\ensuremath{\\mathrm{d}} t+C\\int_{h^2+\\ensuremath{\\Delta t}}^Tt^{-\\frac{1-\\beta}2}(h^2+\\ensuremath{\\Delta t})^\\beta t^{-\\frac{1+\\beta}2}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&= C(h^2+\\ensuremath{\\Delta t})^\\beta(1+|\\log(h^2+\\ensuremath{\\Delta t})|)\\\\\n&\\leq C(h^{2\\beta}+(\\ensuremath{\\Delta t})^\\beta)|\\log(h^2+\\ensuremath{\\Delta t})|.\n\\end{aligned}\n\\end{equation}\nfor $h^2+\\ensuremath{\\Delta t}\\leq 1\/e$, where $C>0$ depends on $T$.\nThe combination of \\eqref{eq:proofSHE1}, \\eqref{eq:proofSHE2}, \\eqref{eq:proofSHE3} and \\eqref{eq:proofSHE4} finishes the proof.\n\\end{proof}\n\\subsection{Stochastic Volterra integro-differential equations}\\label{subsec:sve}\nHere we consider a stochastic integro-differential equation of Volterra-type where the deterministic equation exhibits a parabolic character. The error analysis is basically analogous to the heat equation and therefore we skip some computational details.\n\\label{rem:volterra} The weak solution of the Volterra-type stochastic evolution equation, a simple model of viscoelastic materials in the presence of noise,\n$$\n\\ensuremath{\\mathrm{d}} X(t) + \\left ( \\int_0^t \\frac{1}{\\Gamma(\\rho-1)}(t-s)^{\\rho-2} \\Lambda X(s) \\, \\ensuremath{\\mathrm{d}} s \\right ) \\, \\ensuremath{\\mathrm{d}} t = \\ensuremath{\\mathrm{d}} L(t),~t\\in (0,T]; ~ X(0) =X_0\\in H,\n$$\nis also given by \\eqref{eq:X}, where $(E(t))_{t\\ge 0}$ is the solution operator of the linear, homogeneous deterministic problem (see, for example, \\cite{BGK15,CDaPP,Sperlich} for Gaussian and fractional Brownian noise) and $\\rho\\in (1,2)$. Because of the special convolution kernel above, the equation can be also viewed as a fractional-in-time differential equation.\nUsing the same finite element approximation in space as for the heat equation and a convolution quadrature in time we consider the following recurrence (see \\cite{KovPri14a,KovPri14b} for the Gaussian case),\n\\begin{equation} \\label{eq:full_scheme volterra}\nX^n_{h,\\Delta t} - X^{n-1}_{h,\\Delta t} + \\Delta t \\left ( \\sum_{k=1}^{n} \\omega_{n-k}\\,  \\Lambda_{h} X^k_{h,\\Delta t} \\right ) = P_h(L(t_n)-L(t_{n-1})), \\quad n\\geq 1,\n\\end{equation}\nwith $X^0_{h,\\Delta t}=P_hX_0$ and convolution weights $(\\omega_k)_{k\\ge 0}$ chosen according to (see \\cite{lubich88,lubich88II})\n\\begin{equation} \\label{eq:weight}\n\\left ( \\frac{1 - z}{\\Delta t} \\right )^{1-\\rho} = \\sum_{k\\geq 0} \\omega_k z^k, \\quad |z|<1.\n\\end{equation}\nThe solution to \\eqref{eq:full_scheme volterra} can again be written in the form \\eqref{eq:schemeSHE2} with a suitable operator family $(E^n_{h,\\Delta t})_{n\\in \\mathbb{N}}$, (see \\cite{KovPri14a,KovPri14b} for the Gaussian case). Then, define $(\\tilde{E}(t))_{t\\in [0,T]}$ and $(\\tilde{X}(t))_{t\\in [0,T]}$ according to \\eqref{eq:EtildeSHE} and \\eqref{eq:XtildeSHE}, respectively. In contrast to the heat equation where \\eqref{eq:detEstSHE1} and \\eqref{eq:detEstSHE2} hold, one has the deterministic estimates (see \\cite{Lubich_et_al1996} and \\cite[Theorem 3.1]{KovPri14b})\n\\begin{align}\n\\nnrm{\\tilde E(s)-E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C(h^{2\/\\rho}+\\ensuremath{\\Delta t})s^{-1},\\label{eq:detEstSVE1}\\\\\n\\nnrm{\\Lambda^\\alpha E(s)}{\\ensuremath{\\mathscr L}(H)}+\\nnrm{\\Lambda^\\alpha \\tilde E(s)}{\\ensuremath{\\mathscr L}(H)}&\\leq C s^{-\\rho\\alpha},\\quad 0\\leq\\alpha\\leq 1\/{(2\\rho)},\\label{eq:detEstSVE2}\n\\end{align}\n$s\\in(0,T]$, where $C>0$ does not depend on $h$, $\\ensuremath{\\Delta t}$ and $s$. Note further, that if $\\nnrm{\\Lambda^{-1\/(2\\rho)}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}( H)}<\\infty$, then using It\\^o's isometry, we get the stability estimate as in the Gaussian case, see \\cite[page 2333]{KovPri14a},\n$$\n\\nnrm{X(T)}{L^2(\\Omega,H)}\\le C (\\nnrm{\\Lambda^{-1\/(2\\rho)}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}( H)}+\\nnrm{X_0}{L^2(\\Omega,H)}),\n$$\nand, in particular, Assumption \\ref{ass:abstractSetting} (iv) holds. Furthermore, for $t\\in[\\epsilon, T]$ and $x\\in H$, we have that\n\\begin{equation*}\n\\|E(t)x\\|_{H}=\\|\\Lambda^{1\/(2\\rho)}E(t)\\Lambda^{-1\/(2\\rho)}x\\|_H\\le \\|\\Lambda^{1\/(2\\rho)}E(t)\\|_{\\ensuremath{\\mathscr L}(H)}\\|\\,\\|\\Lambda^{-1\/(2\\rho)}x\\|_{H}\\le \\|\\Phi_{\\epsilon}x\\|_H.\n\\end{equation*}\nTherefore, Assumption \\ref{ass:abstractSetting} (v) holds with $\\Phi_{\\epsilon}:=\\sup_{t\\in [\\epsilon, T]}\\|\\Lambda^{1\/(2\\rho)}E(t)\\|_{\\mathcal{L}(H)}\\Lambda^{-1\/(2\\rho)}$, where the supremum is finite because of \\eqref{eq:detEstSVE2}.\nHence, via an analogous calculation as for the heat equation above, setting $H=U=L^2(\\ensuremath{\\mathcal{O}})$, $B=\\id H$, assuming $\\|\\Lambda^{\\frac{\\beta-1\/\\rho}{2}}Q^\\frac12\\|^2_{\\ensuremath{\\mathscr L_2}(H)}<\\infty$, $\\beta\\in (0,1\/\\rho)$, $X_0\\in L^2(\\Omega,\\cF_0,\\bP;H)$, and using \\eqref{eq:detEstSVE1} and \\eqref{eq:detEstSVE2}, one arrives at the weak error estimate\n\\[\\big|\\bE\\big(g(X^N_{h,\\ensuremath{\\Delta t}})-g(X(T))\\big)\\big|\\leq C(h^{2\\beta}+(\\ensuremath{\\Delta t})^{\\rho\\beta})|\\log(h^{2\/\\rho}+\\ensuremath{\\Delta t})|, \\]\nfor $h^{2\/\\rho}+ \\ensuremath{\\Delta t}\\leq 1\/e$. This is essentially twice the strong rate where the latter is the same as in the Gaussian case \\cite{KovPri14a}, as the strong error analysis carries over to our setting, cf.~Remark \\ref{rem:strongErrorSHE}.\n\n\n\\section{Application to the wave equation}\n\n\\label{sec:Wave}\n\nHere, we apply the general error representation from Section~\\ref{sec:An error representation formula} to a discretization of the stochastic wave equation~\\eqref{eq:SWE}.\n\nLet $\\ensuremath{\\mathcal{O}}\\subset\\bR^d$ be a convex bounded domain and let the spaces $\\dot H^\\alpha$, $\\alpha\\in\\bR$, be as in Section~\\ref{sec:Heat}. We use the product spaces\n\\begin{equation*}\n\\cH^\\alpha:=\\dot H^\\alpha\\times\\dot H^{\\alpha-1},\\quad\\alpha\\in\\bR,\n\\end{equation*}\nwith inner product $\\langle v,w\\rangle_{\\cH^\\alpha}:=\\langle v_1,w_1\\rangle_{\\alpha}+\\langle v_2,w_2\\rangle_{\\alpha-1}$, $v=(v_1,v_2)^\\top$, $w=(w_1,w_2)^\\top$ and norm $\\nnrm{v}{\\cH^\\alpha}=(|v_1|_{\\alpha}^2+|v_2|_{\\alpha-1}^2)^{1\/2}$, where $\\langle\\cdot,\\cdot\\rangle_\\alpha$ and  $\\langle\\cdot,\\cdot\\rangle_{\\alpha-1}$ are the inner products in $\\dot H^\\alpha$ and $\\dot H^{\\alpha-1}$ corresponding to the norms $|\\cdot|_\\alpha$ and $|\\cdot|_{\\alpha-1}$ introduced in Section~\\ref{sec:Heat}. We set\n\\[ H:=\\cH^0=\\dot H^0\\times\\dot H^{-1}=L^2(\\ensuremath{\\mathcal{O}})\\times H^{-1}(\\ensuremath{\\mathcal{O}}),\\quad  U:=\\dot H^0=L^2(\\ensuremath{\\mathcal{O}})\\] and define operators $A:D(A)\\subset  H\\to H$ and $B\\in\\ensuremath{\\mathscr L}( U, H)$ by setting $D(A):= \\cH^1$ and\n\\begin{equation*}\nA:=\\begin{pmatrix}0 & -I \\\\ \\Lambda & 0\\end{pmatrix},\\quad B:=\\begin{pmatrix}0\\\\I\\end{pmatrix},\n\\end{equation*}\nwhere the Laplace operator\n$\\Lambda$ from Section~\\ref{sec:Heat} is now considered as an operator from $\\dot H^1$ to $\\dot H^{-1}$. It is well-known that $-A$ generates a strongly continuous semigroup $(E(t))_{t\\geq0}\\subset\\ensuremath{\\mathscr L}( H)$ given by\n\\begin{equation}\\label{eq:E(t)SWE}\nE(t)=\\begin{pmatrix}C(t) & \\Lambda^{-1\/2}S(t) \\\\ -\\Lambda^{1\/2}S(t) & C(t) \\end{pmatrix},\n\\end{equation}\nwhere $C(t):=\\cos(t\\Lambda^{1\/2})$ and $S(t):=\\sin(t\\Lambda^{1\/2})$ are the cosine and sine operators; compare \\cite[Example~B.1]{PesZab07}, \\cite[Section~A.5.4]{DPZ92} and \\cite[Section~3.14]{ABHN11}.\n\nWith these definitions the abstract equation \\eqref{eq:mainEq} becomes the stochastic wave equation~\\eqref{eq:SWE} with $ H$-valued solution $(X(t))_{t\\geq0}=((X_1(t),X_2(t))^\\top)_{t\\geq0}$. As in the Gaussian case, cf.~\\cite[Lemma~4.1]{KovLarLin13}, one sees that the condition $\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$ implies $\\eqref{eq:assEB1}$ and hence the existence of a unique weak solution $X=(X(t))_{t\\geq 0}$, given that the initial condition $X_0=(X_{0,1},X_{0,2})^\\top$ is $ H$-valued and $\\cF_0$-measurable. Furthermore,\n\\begin{equation}\\label{eq:wstab}\n\\nnrm{X(T)}{L^2(\\Omega,H)}\\le T\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}+\\nnrm{X_0}{L^2(\\Omega,H)}.\n\\end{equation}\n\nThe discretization of Eq.~\\eqref{eq:SWE} is done via finite elements\nin space and an $I$-stable rational single step scheme of order in time. (By `$I$-stable' we mean what is called `$I$-acceptable' in \\cite{NorWan79}.) We use the finite element setting introduced in Section~\\ref{sec:Heat}, the only difference being that we assume a possibly higher order approximation property of the Ritz projection $\\Pi_h$ of the form\n\\begin{equation}\\label{eq:basicFEMestimate2}\n\\nnrm{\\Pi_hv-v}{L^2(\\ensuremath{\\mathcal{O}})}\\leq C h^\\beta|v|_\\beta,\\quad v\\in\\dot H^\\beta,\\;1\\leq\\beta\\leq r,\n\\end{equation}\nwith $r\\ge 2$. For $r=2$ this requires no further assumption on the domain $\\ensuremath{\\mathcal{O}}$ (other than convexity) and $S_h$ can be chosen to be the space of continuous piecewise linear functions on a triangulation of $\\ensuremath{\\mathcal{O}}$ with maximal mesh-size $h$, as in the case of the heat equation. For $r>2$ this firstly requires extra assumptions on the domain $\\ensuremath{\\mathcal{O}}$, namely small enough interior angles in case of a polygon, or smooth enough boundary in case of a curved boundary. The reason is that to achieve $r>2$ one needs an elliptic regularity estimate $\\|u\\|_{H^r}\\le C\\|\\Lambda u\\|_{H^{r-2}}$ with $r>2$ instead of \\eqref{eq:ellreg} corresponding to $r=2$. Furthermore, if the boundary is curved, then the triangulation is not exact and one has to be more precise while approximating near the boundary and hence one needs special elements. For example, \\eqref{eq:basicFEMestimate2} holds for $r=4$ for bounded convex domains with smooth boundary and $S_h$ consisting of continuous piecewise cubic polynomials using special, so-called isoparametric elements, near the boundary, see \\cite[Chapter 1]{Tho06}.\nAlthough \\eqref{eq:basicFEMestimate2} does not appear explicitly in the present paper, it is the key ingredient in the proof of the deterministic error estimate for the finite element approximation of the wave equation which we will use later on and hence we state it as our abstract assumption on the finite element spaces $S_h$.\nLet the discretization $A_h:S_h\\times S_h\\to S_h\\times S_h$ of the operator $A:D(A)\\subset H\\to H$ be defined by\n\\begin{equation*}\nA_h:=\\begin{pmatrix}0 & -I \\\\ \\Lambda_h & 0\\end{pmatrix},\n\\end{equation*}\nwhere $\\Lambda_h:S_h\\to S_h$ is the discrete Laplacian introduced in \\eqref{eq:discreteLaplace}.\nThen $-A_h$ generates a strongly continuous semigroup $(E_h(t))_{t\\geq0}\\subset \\ensuremath{\\mathscr L}(S_h\\times S_h)$. As in Section~\\ref{sec:Heat}, we consider for $N\\in\\bN$ a uniform grid $t_n=n\\ensuremath{\\Delta t}=n (T\/N)$, $n=0,\\ldots,N$, on a finite time interval $[0,T]$. We approximate the operators $E_h(t_n)\\in\\ensuremath{\\mathscr L}(S_h\\times S_h)$ by\n\\[E_{h,\\ensuremath{\\Delta t}}^n:=(R(\\ensuremath{\\Delta t} A_h))^n,\\]\nwhere $R$ is a rational function that satisfies the approximation and stability properties\n\\begin{equation*}\n\\begin{aligned}\n|R(iy)-e^{-iy}|&\\leq C |y|^{p+1},\\quad |y|\\leq b,\\\\\n|R(iy)|&\\leq 1,\\quad y\\in\\bR,\n\\end{aligned}\n\\end{equation*}\nfor some positive integer $p$ and some $b>0$; see \\cite{BakBra79, BreTho80} for details. For instance, choosing $R(\\lambda)=1\/(1-\\lambda)$ and  $R(\\lambda)=(2-\\lambda)\/(2+\\lambda)$ yields the backward Euler method ($p=1$) and the Crank-Nicolson method ($p=2$), respectively.\n\nThe numerical scheme for the stochastic wave equation \\eqref{eq:mainEq} can now be formulated as follows:\nFor $h>0$ and $N\\in\\bN$, the discretization $(X^n_{h,\\ensuremath{\\Delta t}})_{n=0,\\ldots,N}$ of $(X(t))_{t\\in[0,T]}$ in space and time is given as the solution to\n\\begin{equation}\\label{eq:schemeSWE1}\nX_{h,\\ensuremath{\\Delta t}}^n=E_{h,\\ensuremath{\\Delta t}}\\big(X_{h,\\ensuremath{\\Delta t}}^{n-1}+P_h B(L(t_n)-L(t_{n-1}))\\big),\\quad n=1,\\ldots,N;\\quad X_{h,\\ensuremath{\\Delta t}}^0=P_hX_0.\n\\end{equation}\nBy slight abuse of notation, we denote here and in the sequel by $P_h$ both the generalized $L^2$-projection from $\\dot H^{-1}$ onto $S_h$ defined by $\\langle P_hv,w\\rangle_{L^2(\\ensuremath{\\mathcal{O}})}=\\langle v,w\\rangle_{\\dot H^{-1}\\times\\dot H^1}$, $v\\in \\dot H^{-1}$, $w\\in S_h$, and the corresponding projection from $ H=\\dot H^{0}\\times\\dot H^{-1}$ onto $S_h\\times S_h$ defined by the action of the former projection on the coordinates of elements in $\\dot H^{0}\\times\\dot H^{-1}$. Moreover, $P^1: H\\to\\dot H^0$ is the projection of elements in $H=\\dot H^0\\times\\dot H^{-1}$ on the first coordinate.\n\n\\begin{remark}[strong error]\\label{rem:strongErrorSWE}\nAs observed for the discretization of the heat equation in Remark~\\ref{rem:strongErrorSHE}, strong $L^2$-error estimates for the scheme \\eqref{eq:schemeSWE1} carry over from the Gaussian case in the L\\'{e}vy $L^2$-martingale case since they only use It\\^{o}'s isometry \\eqref{eq:ItoIsom}. Arguing as in the proof of \\cite[Theorem~4.13]{KovLarLin13}, we obtain that, if\n\\begin{equation}\\label{eq:SWEassQ}\n\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty\\quad\\text{ and }\\quad X_0\\in L^2(\\Omega,\\cF_0,\\bP; \\cH^{\\beta})\n\\end{equation}\nfor some $\\beta>0$, then the scheme \\eqref{eq:schemeSWE1} approximates the first component $X_1=P^1X$ of the solution $X$ to \\eqref{eq:mainEq} with strong order $\\min(\\beta r\/(r+1),r)$ in space and $\\min(\\beta p\/(p+1),1)$ in time:\n\\[\\nnrm{X^n_{h,\\ensuremath{\\Delta t},1}-X_1(t_n)}{L^2(\\Omega;\\cdot H^0)}\\leq C\\big(h^{\\min(\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\beta\\frac{p}{p+1},1)}\\big),\\quad n=0,\\ldots,N.\\]\nHere we have set $X^n_{h,\\ensuremath{\\Delta t},1}:=P^1X^n_{h,\\ensuremath{\\Delta t}}$. The condition \\eqref{eq:SWEassQ} implies that the solution $X=(X(t))_{t\\geq0}$ takes values in $ \\cH^\\beta$, cf.~\\cite[Theorem~3.1]{KovLarSae10}.\n\n\\end{remark}\n\nThe solution to the scheme \\eqref{eq:schemeSWE1} is given by\n\\begin{equation*}\\label{eq:schemeSWE2}\nX^n_{h,\\ensuremath{\\Delta t}}=E_{h,\\ensuremath{\\Delta t}}^nP_hX_0+\\sum_{j=1}^n E^{n-j+1}_{h,\\ensuremath{\\Delta t}}P_hB(L(t_n)-L(t_{n-1})),\\quad n=0,\\ldots,N.\n\\end{equation*}\nFor $t\\in[0,T]$, define operators $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)\\in\\ensuremath{\\mathscr L}( H)$ by\n\\begin{equation}\\label{eq:EtildeSWE}\n\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t):=\\mathds 1_{\\{0\\}}(t)P_h+\\sum_{j=1}^N\\mathds 1_{(t_{n-1},t_n]}(t)E_{h,\\ensuremath{\\Delta t}}^nP_h,\n\\end{equation}\nwhere the projection $P_h$ is understood as a mapping from $ H=\\dot H^{0}\\times\\dot H^{-1}$ to $S_h\\times S_h$. Then, analogously to the corresponding argument in Section~\\ref{sec:Heat}, one sees that the $S_h\\times S_h$-valued process $(\\tilde X(t))_{t\\in[0,T]}=(\\tilde X_{h,\\ensuremath{\\Delta t}}(t))_{t\\in[0,T]}$ defined by\n\\begin{equation}\\label{eq:XtildeSWE}\n\\tilde X(t)=\\tilde X_{h,\\ensuremath{\\Delta t}}(t):=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)X_0+\\int_0^t\\tilde E_{h,\\ensuremath{\\Delta t}}(t-s)B\\,\\ensuremath{\\mathrm{d}} L(s)\n\\end{equation}\nsatisfies $X^n_{h,\\ensuremath{\\Delta t}}=\\tilde X(t_n)$ $\\bP$-almost surely.\n\nThe proof of the deterministic error estimate in the next lemma is postponed to the end of this section.\n\\begin{lemma} \\label{lem:SWE}\nLet $\\alpha\\geq0$. The operators $E(t)$ and $\\tilde E(t)=\\tilde E_{h,\\ensuremath{\\Delta t}}(t)$ defined in \\eqref{eq:E(t)SWE} and \\eqref{eq:EtildeSWE} satisfy the error estimate\n\\begin{equation}\\label{eq:detEstSWE}\n\\begin{gathered}\n    \\sup_{t\\in[0,T]}\\big(\\nnrm{P^1(\\tilde E(t)-E(t))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\n    +\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\big)\\\\\n    \\leq C\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},1)}\\big),\n\\end{gathered}\n\\end{equation}\nfor $\\ensuremath{\\Delta t}\\leq1$, where $C=C(T)>0$ does not depend on $h$ and $\\ensuremath{\\Delta t}$\n\\end{lemma}\n\nWe are now in the position to prove the following result concerning the weak error of the approximation $X^N_{h,\\ensuremath{\\Delta t},1}:=P^1X^N_{h,\\ensuremath{\\Delta t}}$ of the first component $X_1(T)=P^1X(T)$ of the solution to the stochastic wave eqation~\\eqref{eq:mainEq} at time $T$.\n\n\\begin{theorem}\\label{thm:SWE}\nLet $X_0\\in L^2(\\Omega,\\cF_0,\\bP; \\cH^{2\\beta})$ for some $\\beta>0$ and $g\\in C^2(H,\\bR)$ with $\\sup_{x\\in H}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(H)}<\\infty$. Suppose that either of the following conditions holds.\n\\begin{itemize}\n\\item[\\upshape (i)] $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$ and\n\\begin{equation}\\label{eq:assgSWE}\n\\sup_{x\\in\\dot H^0}\\nnrm{\\Lambda^{\\frac\\beta2}g''(x)\\Lambda^{-\\frac\\beta2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}<\\infty.\n\\end{equation}\n\\item[\\upshape (ii)]\n\\begin{equation}\\label{eq:weqii}\n\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{-\\frac12}p_m y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac{1}{2}}p_m y\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty,\n\\end{equation}\nwhere $p_m$ denotes the orthogonal projection from $\\dot H^0$ to $\\operatorname{span}\\{\\varphi_1,\\ldots,\\varphi_m\\}$, $(\\varphi_k)_{k\\in\\mathbb N}$ being an orthonormal basis of $\\dot H^0$ consisting of eigenvectors of $\\Lambda$.\n\\end{itemize}\nThen, there is a unique weak solution $(X(t))_{t\\geq0}$ to Eq.~\\eqref{eq:mainEq} given by \\eqref{eq:X}. \n\nLet $(X_{h,\\ensuremath{\\Delta t}}^n)_{n=0,\\ldots,N}$ be given by the scheme \\eqref{eq:schemeSWE1}. Then, there exists a constant $C=C(g,T)>0$ that does not depend on $h$ and $\\ensuremath{\\Delta t}$, such that for $\\ensuremath{\\Delta t}\\leq 1$\n$$\n    \\big|\\bE\\big(g(X^N_{h,\\ensuremath{\\Delta t},1})-g(X_1(T))\\big)\\big|\\leq C\\big(h^{\\min(2\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\beta\\frac{p}{p+1},1)}\\big)\n$$\n\\end{theorem}\n\nBefore proving Theorem~\\ref{thm:SWE}, we state two remarks and discuss some examples where the conditions of Theorem \\ref{thm:SWE}, in particular \\eqref{eq:assgSWE} and \\eqref{eq:weqii}, are satisfied.\n\n\\begin{remark}\nAs a consequence of Lemma~\\ref{lem:integrandIsomorphism} and the fact that $\\Lambda^\\alpha p_m\\in\\ensuremath{\\mathscr L_2}(\\dot H^0)$ for all $\\alpha\\in\\bR$ and $m\\in\\bN$, the terms $\\Lambda^{-1\/2}p_m y$ and $\\Lambda^{\\beta-1\/2}p_m y$ in \\eqref{eq:weqii} are defined in an $L^2(U_1,\\nu(\\ensuremath{\\mathrm{d}} y);\\dot H^0)$-sense. The sequence $\\big(\\|\\Lambda^{-1\/2}p_m y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_m y\\|_{\\dot{H}^0}\\big)_{m\\in\\bN}$ is monotonically increasing for $\\nu$-almost all $y\\in U_1$, so that the limit in \\eqref{eq:weqii} is in fact a supremum. Moreover, if we explicitly choose $U_1=\\dot H^{\\beta-1}$ as the state space of $L$, then the condition~(ii) is equivalent to assuming that $\\operatorname{supp}\\nu\\subset \\dot H^{2\\beta-1}$ and\n\\[\\int_{\\dot H^{2\\beta-1}}\\|\\Lambda^{-\\frac12}y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac{1}{2}}y\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty .\\]\nThis choice of $U_1$ is possible w.l.o.g.\\ whenever $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$, since then the natural embedding of $U_0=Q^{1\/2}U=Q^{1\/2}\\dot H^0$ into $\\dot H^{\\beta-1}$ is Hilbert-Schmidt and we can re-expand $L$ in the form \\eqref{eq:LPexpansion} as an $\\dot H^{\\beta-1}$-valued martingale, compare Remark~\\ref{rem:approxL}. However, in the spirit of, e.g., \\cite{AppRie10,Rie12,Rie14}, we prefer a formulation of our results that is independent of the specific choice of the state space $U_1$.\n\\end{remark}\n\n\\begin{remark}\nInstead of the symmetric condition $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$, the sufficient asymmetric condition $\\nnrm{\\Lambda^{\\beta-1\/2}Q\\Lambda^{-1\/2}}{\\ensuremath{\\mathscr L_1}(\\dot{H}^0)}<\\infty$ is imposed in \\cite{KovLarLin13} in the Wiener case in order to double the rate of strong convergence for the wave equation.\nThe asymmetric condition \\eqref{eq:weqii}\nappearing in (ii) above, which is again sufficient for\\linebreak $\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$,\nresembles the same situation in the present case.\n\\end{remark}\n\n\n\\begin{example}\nAn important basic example which satisfies \\eqref{eq:assgSWE} is $g(x)=\\|x\\|^2_{\\dot{H}^0}$.\n\\end{example}\n\n\\begin{example}\nAs another example for a test function $g$ satisfying \\eqref{eq:assgSWE} consider\n\\begin{equation*}\ng(x):=f(\\langle\\varphi_1,x\\rangle_{\\dot H^0},\\ldots,\\langle\\varphi_n,x\\rangle_{\\dot H^0}),\\quad x\\in \\dot H^0,\n\\end{equation*}\nwhere $f\\in C^2(\\bR^n,\\bR)$ has bounded second order derivatives and $(\\varphi_k)_{k\\in\\bN}\\subset D(\\Lambda)$ is an orthonormal basis of $\\dot H^0=L^2(\\ensuremath{\\mathcal{O}})$ consisting of eigenfunctions of $\\Lambda $ with corresponding eigenvalues $(\\lambda_k)_{k\\in\\bN}\\subset(0,\\infty)$. Then, for $x,y\\in\\dot H^0$,\n\\[\n\\Lambda^{\\beta\/2}g''(x)\\Lambda^{-\\beta\/2}y=\\sum_{j,k=1}^n\\lambda_j^{-\\beta\/2}\\lambda_k^{\\beta\/2}(\\partial_j\\partial_kf)\\big(\\langle\\varphi_1,x\\rangle_{\\dot H^0},\\ldots,\\langle\\varphi_n,x\\rangle_{\\dot H^0}\\big)\\langle\\varphi_j,y\\rangle_{\\dot H^0}\\varphi_k\n\\]\nand \\eqref{eq:assgSWE} holds. More generally, the condition \\eqref{eq:assgSWE} is satisfied by all $g\\in C^2(\\dot H^0,\\bR)$ of the form $g=\\tilde g\\circ\\Lambda^{-\\beta\/2}$ with $\\tilde g\\in C^2(\\dot H^0,\\bR)$ satisfying $\\sup_{x\\in\\dot H^0}\\nnrm{\\tilde g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)}<\\infty$. For such $g$ we have $g''(x)=\\Lambda^{-\\beta\/2}\\tilde g''(\\Lambda^{-\\beta\/2}x)\\Lambda^{-\\beta\/2}$.\n\\end{example}\n\n\\begin{example}\\label{ex:spec}\nConsider the situation of Example \\ref{ex:1dlevy}; that is when\n\\[\\nu=\\sum_{k\\in\\bN}\\nu_k\\circ\\pi_k^{-1},\\]\nwhere $\\nu_k$ is the L\\'{e}vy measure of $L_k$ and $\\pi_k:\\bR\\to U_1$ is defined by $\\pi_k(\\xi):=\\xi e_k$. Let us assume that $e_k=\\sqrt{q_k}\\,\\varphi_k$, where $(q_k)_{k\\in\\bN}$ is a bounded sequence of positive numbers and $(\\varphi_k)_{k\\in \\bN}$ is an orthonormal basis of $\\dot{H}^0$ consisting of eigenfunctions of $\\Lambda$ with corresponding eigenvalues $(\\lambda_k)_{k\\in \\bN}$.\nThen, with $p_m$ as in (ii) in Theorem \\ref{thm:SWE},\n\\begin{align*}\n&\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{-\\frac12}p_m y\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac{1}{2}}p_m y\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&=\\sum_{k\\in \\bN}\\int_{\\bR}\\xi^2q_k\\|\\Lambda^{-\\frac12}\\varphi_k\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-\\frac12}\\varphi_k\\|_{\\dot{H}^0}\\,\\nu_k(\\ensuremath{\\mathrm{d}} \\xi)\\\\\n&=\\sum_{k\\in \\bN}\\int_{\\bR}\\xi^2q_k\\lambda_k^{-\\frac12}\\|\\varphi_k\\|_{\\dot{H}^0}\\lambda_k^{\\beta-\\frac12}\\|\\varphi_k\\|_{\\dot{H}^0}\\,\\nu_k(\\ensuremath{\\mathrm{d}} \\xi)\\\\\n&=\\sum_{k\\in \\bN}\\int_{\\bR}\\xi^2q_k\\|\\Lambda^{\\frac{\\beta-1}2}\\varphi_k\\|_{\\dot{H}^0}\\|\\Lambda^{\\frac{\\beta-1}2}\\varphi_k\\|_{\\dot{H}^0}\\,\\nu_k(\\ensuremath{\\mathrm{d}} \\xi)\\\\\n&=\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{\\frac{\\beta-1}2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\frac{\\beta-1}2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\n\\,=\\,\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2,\n\\end{align*}\nwhere we used Lemma~\\ref{lem:integrandIsomorphism} in the last step.\nThat is, when $\\nu$ is concentrated on the eigenspaces $\\{r\\varphi_k:r\\in\\bR\\}$, $k\\in\\bN$, of $\\Lambda$, then the abstract asymmetric condition \\eqref{eq:weqii} coincides with the familiar symmetric Hilbert-Schmidt condition. The situation is similar in the Wiener case \\cite{KovLarLin13} when $\\Lambda$ and $Q$ commute.\n\\end{example}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:SWE}]\nFirst suppose that (i) holds. We apply Theorem~\\ref{thm:errorRep} and Corollary~\\ref{cor:errorRep} with $G=g\\circ P^1$. Note that $G'(x)=(P^1)^*g'(P^1x)\\in H$ and $$G''(x)=(P^1)^*g''(P^1x)P^1\\in\\ensuremath{\\mathscr L}( H)$$ for all $x\\in H$, where $(P^1)^*\\in\\ensuremath{\\mathscr L}(\\dot H^0, H)$ is the Hilbert space adjoint of $P^1\\in\\ensuremath{\\mathscr L}( H,\\dot H^0)$. Using \\eqref{eq:u_xu_xx}\none obtains\n\\begin{equation}\\label{eq:proofSWE0}\nu_x(t,\\xi)=\\bE\\big((P^1)^*g'(P^1Z(T,t,x))\\big)\\big|_{x=\\xi},\\quad u_{xx}(t,\\xi)=\\bE\\big((P^1)^*g''(P^1Z(T,t,x))P^1\\big)\\big|_{x=\\xi}\n\\end{equation}\nfor all $H$-valued random variables $\\xi$ and $t\\in[0,T]$.\n\n\nWe combine \\eqref{eq:wstab}, \\eqref{eq:proofSWE0} and the deterministic error estimate \\eqref{eq:detEstSWE} with $\\alpha=2\\beta$ in order to estimate the first term on the right hand side of the error representation formula~\\eqref{eq:errorRep4} in Corollary~\\ref{cor:errorRep}. We have, where the first inequality follows similarly as for the stochastic heat equation, that\n\\begin{equation}\\label{eq:proofSWE1}\n\\begin{aligned}\n\\big|\\bE\\big\\{&u(0,\\tilde E(T)X_0)-u(0,E(T)X_0)\\big\\}\\big|\\\\\n&\\leq \\int_0^1\\big\\|g'\\big(P^1Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big)\\big\\|_{L^2(\\Omega, \\dot{H}^0)}\\,\\ensuremath{\\mathrm{d}}\\theta\\\\\n&\\qquad \\times \\|P^1(\\tilde E(T)-E(T))X_0\\|_{L^2(\\Omega, \\dot{H}^0)}\\\\\n&\\leq C\\big(1+\\int_0^1\\big\\|Z\\big(T,0,Y(0)+\\theta(\\tilde Y(0)-Y(0))\\big)\\big\\|_{L^2(\\Omega, H)}\\,\\ensuremath{\\mathrm{d}} \\theta\\big)\\\\\n&\\qquad \\times \\nnrm{P^1(\\tilde E(T)-E(T))}{\\ensuremath{\\mathscr L}( \\cH^{2\\beta},\\dot H^0)}\\,\\nnrm{X_0}{L^2(\\Omega; \\cH^{2\\beta})}\\\\\n&\\leq C\\big(1+\\nnrm{\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}+\\nnrm{X_0}{L^2(\\Omega; H)}\\big)\\,\\nnrm{X_0}{L^2(\\Omega; \\cH^{2\\beta})}\\\\\n&\\qquad \\times \\big(h^{\\min(2\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\beta\\frac{p}{p+1},1)}\\big).\n\\end{aligned}\n\\end{equation}\nUsing \\eqref{eq:proofSWE0}, Lemma~\\ref{lem:integrandIsomorphism} and Remark~\\ref{not:Phix}, the integral of the function $\\Psi_1$ in the second term on the right hand side of the formula \\eqref{eq:errorRep4} can be treated as follows:\n\\begin{equation}\\label{eq:proofSWE2}\n\\begin{aligned}\n\\Big|\\bE&\\int_0^T\\int_{ U_1}\\int_0^1\\Psi_1(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{ U_1}\\int_0^1(1-\\theta)\\Big\\langle\\bE\\big(g''\\big(P^1Z(T,t,x+E(T-t)By+\\theta F(T-t)y)\\big)\\big)\\big|_{x=\\tilde Y(t)}\\\\\n&\\quad\\times P^1F(T-t)y,\\;P^1F(T-t)y\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\int_0^T\\nnrm{P^1F(T-t)}{\\ensuremath{\\mathscr L_2}( U_0,\\dot H^0)}^2\\ensuremath{\\mathrm{d}} t\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)} C \\big(h^{\\min(\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\beta\\frac{p}{p+1},1)}\\big)^2\n\\end{aligned}.\n\\end{equation}\nThe last step in \\eqref{eq:proofSWE2} is due to the fact that, by It\\^{o}'s isometry \\eqref{eq:ItoIsom}, the integral in the penultimate line is the square of the strong error $\\nnrm{X^N_{h,k,1}-X_1(T)}{L^2(\\Omega;\\dot H^0)}$ for zero initial condition $X_0=0$; it can be estimated as in the Gaussian case \\cite[Theorem~4.13]{KovLarLin13}, compare Remark~\\ref{rem:strongErrorSWE}.\n\nConcerning the integral of the function $\\Psi_2$ in the second term on the right hand side of Eq.~\\eqref{eq:errorRep4}, we have by \\eqref{eq:proofSWE0}, Lemma~\\ref{lem:integrandIsomorphism}, \\eqref{eq:estHS} and since $U_0=Q^{1\/2}(U)=Q^{1\/2}(\\dot H^0)$,\n\\begin{equation}\\label{eq:proofSWE3}\n\\begin{aligned}\n\\Big|\\bE&\\int_0^T\\int_{ U_1}\\int_0^1\\Psi_2(t,\\theta,y)\\,\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{ U_1}\\int_0^1\\Big\\langle\\bE\\big(g''\\big(P^1Z(T,t,x+\\theta E(T-t)By)\\big)\\big)\\big|_{x=\\tilde Y(t)}\\\\\n&\\quad\\times P^1E(T-t)By,P^1F(T-t)y\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{ U_1}\\int_0^1\\Big\\langle\\bE\\big(\\Lambda^{\\frac\\beta2}g''\\big(P^1Z(T,t,x+\\theta E(T-t)By)\\big)\\Lambda^{-\\frac\\beta2}\\big)\\big|_{x=\\tilde Y(t)}\\\\\n&\\quad\\times \\Lambda^{\\frac\\beta2}P^1E(T-t)B\\Lambda^{\\frac{1-\\beta}2}\\Lambda^{\\frac{\\beta-1}2}y,\\;\\Lambda^{-\\frac\\beta2}P^1F(T-t)\\Lambda^{\\frac{1-\\beta}2}\\Lambda^{\\frac{\\beta-1}2}y\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{\\Lambda^{\\frac\\beta2}g''(x)\\Lambda^{-\\frac\\beta2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\nnrm{\\Lambda^{\\frac{\\beta-1}2}Q^{\\frac12}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\\\\\n&\\quad\\times \\int_0^T\\nnrm{\\Lambda^{\\frac\\beta2}P^1E(T-t)B\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\nnrm{\\Lambda^{-\\frac\\beta2}P^1F(T-t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\,\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation}\nNote that, by the definition of $B = (0,I)^\\top$ and $E(t)$ from \\eqref{eq:E(t)SWE} we have\n\\begin{equation}\\label{eq:etb}\n\\nnrm{\\Lambda^{\\frac\\beta2}P^1E(T-t)B\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\n=\\nnrm{\\Lambda^{\\frac{\\beta-1}2}S(T-t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\n=\\nnrm{S(T-t)}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\leq1;\n\\end{equation}\nit remains to estimate the integral\n\\begin{equation*}\\label{eq:proofSWE4}\n\\begin{aligned}\n\\int_0^T\\nnrm{\\Lambda^{-\\frac\\beta2}P^1F(T-t)\\Lambda^{\\frac{1-\\beta}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\ensuremath{\\mathrm{d}} t\n&=\\int_0^T\\nnrm{P^1F(t)}{\\ensuremath{\\mathscr L}(\\dot H^{\\beta-1},\\dot H^{-\\beta})}\\ensuremath{\\mathrm{d}} t\\\\\n&=\\int_0^T\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\beta-1},\\dot H^{-\\beta})}\\ensuremath{\\mathrm{d}} t.\n\\end{aligned}\n\\end{equation*}\nTo this end, it suffices to apply the deterministic error estimate \\eqref{eq:detEstSWE} with $\\alpha=2\\beta$. The combination of \\eqref{eq:proofSWE1}, \\eqref{eq:proofSWE2} and \\eqref{eq:proofSWE3} finishes the proof.\n\nNext, suppose that (ii) holds.\nBy Lemma~\\ref{lem:integrandIsomorphism} we have\n\\begin{comment}\n\\begin{align*}\n&\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n=\\lim_{m\\to\\infty}\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n=\\lim_{m\\to\\infty}\\nnrm{\\Lambda^{(\\beta-1)\/2}p_mQ^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\\\\\n&=\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{(\\beta-1)\/2}p_my\\|_{\\dot{H}^0}^2\\,\\nu(\\ensuremath{\\mathrm{d}} y)\n=\\lim_{m\\to\\infty}\\int_{U_1}\\langle\\Lambda^{-1\/2}p_my,\\Lambda^{\\beta-1\/2}p_my\\rangle_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\lim_{m\\to\\infty}\\int_{U_1}\\|\\Lambda^{-1\/2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)<\\infty,\n\\end{align*}\n\\end{comment}\n\\begin{align*}\n\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n&=\\nnrm{\\Lambda^{(\\beta-1)\/2}p_mQ^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n=\\int_{U_1}\\|\\Lambda^{(\\beta-1)\/2}p_my\\|_{\\dot{H}^0}^2\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&=\\int_{U_1}\\langle\\Lambda^{-1\/2}p_my,\\Lambda^{\\beta-1\/2}p_my\\rangle_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\\\\n&\\leq\\int_{U_1}\\|\\Lambda^{-1\/2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y),\n\\end{align*}\nhence,\n$$\n    \\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n    =\\lim_{m\\to\\infty}\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}^2\n    < \\infty,\n$$\nproving that there is a unique weak solution $(X(t))_{t\\geq0}$ to Eq.~\\eqref{eq:mainEq} given by \\eqref{eq:X}.\nTo estimate the weak error, we apply an approximation procedure and consider for $m\\in\\bN$ the $H$-valued random variables\n$X^{[m]}(T):=E(T)X_0+\\int_0^TE(T-s)Bp_m\\,\\ensuremath{\\mathrm{d}} L(s)$ and $\\tilde X^{[m]}(T)=\\tilde X^{[m]}_{h,\\ensuremath{\\Delta t}}(T):=\\tilde E(T)X_0+\\int_0^T\\tilde E(T-s)Bp_m\\,\\ensuremath{\\mathrm{d}} L(s)$. Using It\\^{o}'s isometry and the fact that $\\nnrm{(I-p_m)\\Lambda^{-1\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}\\to 0$ as $m\\to\\infty$, we get that $X^{[m]}(T)\\xrightarrow{m\\to\\infty} X(T)$ and $\\tilde X^{[m]}(T)\\xrightarrow{m\\to\\infty}\\tilde X(T)$ in $L^2(\\Omega;H)$. As a consequence of this and \\eqref{eq:growthGG'}, we obtain $e^{[m]}(T):=\\bE\\big( G(\\tilde X^{[m]}(T))-G(X^{[m]}(T))\\big)\\xrightarrow{m\\to\\infty}  e(T)$. Thus, it suffices to show the desired decay rate for the error $e^{[m]}(T)$ with a constant that does not depend on $m$. To this end, we observe that the estimate \\eqref{eq:proofSWE1} can be used without any change, and that the analogue to the estimate \\eqref{eq:proofSWE2} gives indeed the desired rate if we use that $\\nnrm{p_m\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}\\leq\\nnrm{\\Lambda^{(\\beta-1)\/2}Q^{1\/2}}{\\ensuremath{\\mathscr L_2}(\\dot H^0)}<\\infty$. Finally, the estimate corresponding to \\eqref{eq:proofSWE3} reads\n\n\n\\begin{equation*}\\label{eq:proofSWE3b}\n\\begin{aligned}\n\\Big|&\\bE\\int_0^T\\int_{U_1}\\int_0^1\\Big\\langle\\bE\\big(g''\\big(P^1Z^{[m]}(T,t,x+\\theta E(T-t)Bp_my)\\big)\\big)\\big|_{x=\\tilde Y^{[m]}(t)}\\\\\n&\\quad\\times P^1E(T-t)Bp_my,P^1F(T-t)p_my\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&= \\Big|\\bE\\int_0^T\\int_{U_1}\\int_0^1\\Big\\langle\\bE\\big(g''\\big(P^1Z^{[m]}(T,t,x+\\theta E(T-t)Bp_my)\\big)\\big)\\big|_{x=\\tilde Y^{[m]}(t)}\\\\\n&\\quad\\times P^1E(T-t)B\\Lambda^{1\/2}\\Lambda^{-1\/2}p_my,P^1F(T-t)\\Lambda^{1\/2-\\beta}\\Lambda^{\\beta-1\/2}p_my\\Big\\rangle_{\\dot H^0}\\ensuremath{\\mathrm{d}}\\theta\\,\\nu(\\ensuremath{\\mathrm{d}} y)\\,\\ensuremath{\\mathrm{d}} t\\Big|\\\\\n&\\leq \\sup_{x\\in\\dot H^0}\\nnrm{g''(x)}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\int_0^T\\nnrm{P^1E(T-t)B\\Lambda^{1\/2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\nnrm{P^1F(T-t)\\Lambda^{1\/2-\\beta}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\,\\ensuremath{\\mathrm{d}} t\\\\\n&\\quad\\times \\int_{U_1}\\|\\Lambda^{-1\/2}p_my\\|_{\\dot{H}^0}\\|\\Lambda^{\\beta-1\/2}p_my\\|_{\\dot{H}^0}\\,\\nu(\\ensuremath{\\mathrm{d}} y),\n\\end{aligned}\n\\end{equation*}\nwhere $Z^{[m]}$ and $\\tilde Y^{[m]}$ are defined by replacing $B$ by $Bp_m$ in the definitions of $Z$ and $\\tilde Y$.\nBy \\eqref{eq:detEstSWE} with $\\alpha=2\\beta$\nand the fact that $\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1}, \\cH^\\alpha)}=1$ we have that\n\\begin{align*}\n&\\sup_{t\\in[0,T]}\\nnrm{P^1F(T-t)\\Lambda^{1\/2-\\beta}}{\\ensuremath{\\mathscr L}(\\dot H^0)}=\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))B\\Lambda^{1\/2-\\beta}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\\\\n&\\quad =\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{2\\beta-1},\\dot H^0)}\\leq C\\big(h^{\\min(2\\beta\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\beta\\frac{p}{p+1},1)}\\big).\n\\end{align*}\nFinally, by \\eqref{eq:weqii} and \\eqref{eq:etb} with $\\beta =0$, the proof is complete.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:SWE}]\nWe use the estimates\n\\begin{equation}\\label{eq:detEstSWE1}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(E^n_{h,\\ensuremath{\\Delta t}}P_h-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:detEstSWE2}\n\\nnrm{E(t)-E(s)}{\\ensuremath{\\mathscr L}( \\cH^\\delta, H)}\\leq C|t-s|^\\delta,\\quad t,\\,s\\geq0,\\;\\delta\\in[0,1].\n\\end{equation}\nfrom Corollary~4.11 and Lemma~4.4 in \\cite{KovLarLin13}. Corollary~4.11 in \\cite{KovLarLin13} is based on an error estimate proved in \\cite{BakBra79}.\n\nBecause of the `piecewise' definition of $\\tilde E(t)$ in \\eqref{eq:EtildeSWE}, the combination of \\eqref{eq:detEstSWE1} and \\eqref{eq:detEstSWE2} gives\n\\begin{equation}\\label{eq:detEstSWE3}\n\\begin{aligned}\n&\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\\\\\n&\\leq \\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}+\\sup_{n\\in\\{1,\\ldots,N\\}}\\sup_{t\\in(t_{n-1},t_n)}\\nnrm{E(t_n)-E(t)}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\cH)}\\\\\n&\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha,1)}\\big)\\\\\n&=C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},1)}\\big)\n\\end{aligned}\n\\end{equation}\nfor $\\ensuremath{\\Delta t}\\leq 1$.\nIt remains to show that\n\\begin{equation}\\label{eq:detEstSWE4}\n\\sup_{t\\in[0,T]}\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},1)}\\big).\n\\end{equation}\nTo this end, we will prove the estimate\n\\begin{equation}\\label{eq:detEstSWE5}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big).\n\\end{equation}\nThen, \\eqref{eq:detEstSWE4} follows from \\eqref{eq:detEstSWE5} and \\eqref{eq:detEstSWE2} by estimating analogously to \\eqref{eq:detEstSWE3} and using the fact that\n\\begin{align*}\n\\nnrm{P^1(E(t_n)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\n&=\\nnrm{\\Lambda^{-\\frac\\alpha4}P^1(E(t_n)-E(t))B\\Lambda^{\\frac12-\\frac\\alpha4}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\\\\n&=\\nnrm{P^1(E(t_n)-E(t))B\\Lambda^{\\frac{1-\\alpha}2}}{\\ensuremath{\\mathscr L}(\\dot H^0)}\\\\\n&\\leq \\nnrm{P^1(E(t_n)-E(t))}{\\ensuremath{\\mathscr L}( \\cH^\\alpha,\\dot H^0)}\\nnrm{B\\Lambda^{\\frac{1-\\alpha}2}}{\\ensuremath{\\mathscr L}(\\dot H^0, \\cH^\\alpha)},\n\\end{align*}\nwhere $\\nnrm{B\\Lambda^{\\frac{1-\\alpha}2}}{\\ensuremath{\\mathscr L}(\\dot H^0, \\cH^\\alpha)}=\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1}, \\cH^\\alpha)}=1$.\n\nIn order to show \\eqref{eq:detEstSWE5}, we distinguish the cases $\\alpha>2$ and $0\\leq\\alpha\\leq2$.\nFor $\\alpha>2$ we have by \\eqref{eq:detEstSWE1}\n\\begin{equation}\\label{eq:proofSWE6}\n\\begin{aligned}\n\\sup_{n\\in\\{0,\\ldots,N\\}}&\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1},\\dot H^0)}\\\\\n&\\leq\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^{\\alpha},\\dot H^0)}\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1}, \\cH^\\alpha)}\\\\\n&\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big)\n\\end{aligned}\n\\end{equation}\nAs the operator $P^1(\\tilde E(t)-E(t))B\\in\\ensuremath{\\mathscr L}(\\dot H^0)$ is symmetric in $\\dot H^0$ and since $\\dot H^{-\\alpha+1}$ can be identified with the dual space of $\\dot H^{\\alpha-1}$, we have\n\\begin{equation*}\n\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1},\\dot H^0)}=\\nnrm{P^1(\\tilde E(t)-E(t))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-\\alpha+1})}\n\\end{equation*}\nand therefore also\n\\begin{equation}\\label{eq:proofSWE7}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-\\alpha+1})}\\leq C(T)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big).\n\\end{equation}\nNext, we use the fact that $\\dot H^{(\\alpha\/2)-1}$ and $\\dot H^{-\\alpha\/2}$ can be represented as the real interpolation spaces $(\\dot H^0,\\dot H^{\\alpha-1})_{\\theta,2}$ and $(\\dot H^{-\\alpha+1},\\dot H^0)_{\\theta,2}$, respectively, where $\\theta=((\\alpha\/2)-1)\/(\\alpha-1)\\in(0,1)$, cf.~Remark~\\ref{rem:interpolation}. Thus, interpolation between \\eqref{eq:proofSWE6} and \\eqref{eq:proofSWE7} yields\n\\begin{equation*}\n\\begin{aligned}\n&\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\\\\n&\\leq\\sup_{n\\in\\{0,\\ldots,N\\}}C(\\alpha)\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-\\alpha+1})}^{1-\\theta}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{\\alpha-1},\\dot H^0)}^\\theta\\\\\n&\\leq C(T,\\alpha)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big),\n\\end{aligned}\n\\end{equation*}\nsee, e.g., Definition 1.2.2\/2 and Theorem 1.3.3(a) in \\cite{Tri78}.\n\nFor $0\\leq\\alpha\\leq2$, we note that\n\\begin{equation*}\n\\begin{aligned}\n\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-1})}\n&=\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{1},\\dot H^0)}\\\\\n&\\leq\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( \\cH^2,\\dot H^0)}\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^1, \\cH^2)},\\\\\n\\end{aligned}\n\\end{equation*}\nwhere we used again the symmetry of $P^1(\\tilde E(t)-E(t))B\\in\\ensuremath{\\mathscr L}(\\dot H^0)$.\nBy \\eqref{eq:detEstSWE1} we obtain\n\\begin{equation}\\label{eq:proofSWE8}\n\\sup_{n\\in\\{0,\\ldots,N\\}}\n\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^0,\\dot H^{-1})}\n\\leq C(T)\\big(h^{\\min(2\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\frac{p}{p+1},p)}\\big),\n\\end{equation}\nwhich is \\eqref{eq:detEstSWE5} for $\\alpha=0$.\nMoreover, also by \\eqref{eq:detEstSWE1},\n\\begin{equation}\\label{eq:proofSWE9}\n\\begin{aligned}\n\\sup_{n\\in\\{0,\\ldots,N\\}}&\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{-1},\\dot H^{0})}\\\\\n&\\leq \\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))}{\\ensuremath{\\mathscr L}( H,\\dot H^{0})}\\nnrm{B}{\\ensuremath{\\mathscr L}(\\dot H^{-1}, H)}\n\\;\\leq\\; C(T),\n\\end{aligned}\n\\end{equation}\ni.e., we have \\eqref{eq:detEstSWE5} for $\\alpha=2$. Finally, if $\\alpha\\in(0,2)$, interpolation with $\\theta=(\\alpha\/2)-1\\in(0,1)$ between \\eqref{eq:proofSWE8} and \\eqref{eq:proofSWE9} gives\n\\begin{align*}\n    &\\sup_{n\\in\\{0,\\ldots,N\\}}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{(\\alpha\/2)-1},\\dot H^{-\\alpha\/2})}\\\\\n    &\\quad\\leq\\sup_{n\\in\\{0,\\ldots,N\\}}C(\\alpha)\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{0},\\dot H^{-1})}^{1-\\theta}\\nnrm{P^1(\\tilde E(t_n)-E(t_n))B}{\\ensuremath{\\mathscr L}(\\dot H^{-1},\\dot H^{0})}^{\\theta}\\\\\n    &\\quad\\leq C(T,\\alpha)\\big(h^{\\min(2\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(2\\frac{p}{p+1},p)}\\big)^{\\frac\\alpha2}\\\\\n    &\\quad= C(T,\\alpha)\\big(h^{2\\frac{r}{r+1}}+(\\ensuremath{\\Delta t})^{2\\frac{p}{p+1}}\\big)^{\\frac\\alpha2}\\\\\n    &\\quad\\leq C(T,\\alpha)\\big(h^{\\min(\\alpha\\frac{r}{r+1},r)}+(\\ensuremath{\\Delta t})^{\\min(\\alpha\\frac{p}{p+1},p)}\\big).\\qedhere\n\\end{align*}\n\\end{proof}\n\n\n\n\n\\begin{appendix}\n\\section{Poisson random measures and a comparison of\\\\ stochastic integrals}\n\\label{sec:PRM+SI}\n\nOur proof of Theorem~\\ref{thm:errorRep} is based on It\\^{o}'s formula for Banach space-valued jump processes driven by Poisson random measures as presented in \\cite{MRT13}. Alternatively, one could use It\\^{o}'s formula\nas proved in \\cite{GraPel74}, but the formula in \\cite{MRT13} is more convenient in our setting.\nIn this section, we use Lemma~\\ref{lem:integrandIsomorphism} to relate our setting to the setting in \\cite{MRT13}.\n\nIt is well-known that the jumps of a L\\'{e}vy process determine a Poisson random measure on the product space of the underlying time interval and the state space. We refer to \\cite[Section~6]{PesZab07} for a definition and properties of Poisson random measures. For $(\\omega,t)\\in\\Omega\\times(0,\\infty)$ we denote by $\\Delta L(t)(\\omega):=L(t)(\\omega)-\\lim_{s\\nearrow t}L(s)(\\omega)\\in U_1$ the jump of a trajectory of $L$ at time $t$. Setting\n\\begin{equation*}\nN(\\omega):=\\sum_{\\Delta L(t)(\\omega)\\neq0}\\delta_{(t,\\Delta L(t)(\\omega))},\\quad \\omega\\in\\Omega,\n\\end{equation*}\ndefines a Poisson random measure $N$ on $([0,\\infty)\\times U_1,\\cB([0,\\infty))\\otimes\\cB(U_1))$ with intensity measure (or compensator) $\\lambda\\otimes\\nu$, where $\\lambda$ is Lebesgue measure on $[0,\\infty)$ and $\\nu$ is the jump intensity measure of $L$.\nThis follows, e.g., from Theorem~6.5 in \\cite{PesZab07} together with Theorems~4.9, 4.15, 4.23 and Lemma~4.25 therein.\nWe denote the compensated Poisson random measure by\n\\begin{equation}\nq:=N-\\lambda\\otimes \\nu.\n\\end{equation}\n\nLet $V$ be a (real and separable) Hilbert space. The stochastic integral with respect to $q$ of functions in $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes \\nu;V)=L^2(\\Omega_T\\times U_1,\\cP_T\\otimes\\cB(U_1),\\bP_T\\otimes \\nu;V)$  is constructed as a linear isometry\n\\begin{equation*\nL^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;V)\\to \\cM^2_T(V),\\;\nf\\mapsto\\Big(\\int_0^t\\int_{U_1} f(s,x)\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\Big)_{t\\in[0,T]}.\n\\end{equation*}\nIn particular, the $V$-valued integral processes have c\\`{a}dl\\`{a}g modifications; we will always work with such a c\\`{a}dl\\`{a}g modification.\nUsing a standard stopping procedure, the stochastic integral can be extendend to functions $f\\in L^0(\\Omega_T\\times U_1,\\cP_T\\otimes\\cB(U_1),\\bP_T\\otimes \\nu;V)$ such that\n\\[\\bP\\Big(\\int_0^T\\int_{U_1}\\nnrm{f(s,x)}V^2\\,\\nu(\\ensuremath{\\mathrm{d}} x)\\,\\ensuremath{\\mathrm{d}} s<\\infty\\Big)=1.\\]\nWe refer to \\cite{MRT13}, \\cite{Pre10} and the references therein for details on stochastic integration w.r.t.\\ Poisson random measures, compare also \\cite[Section~8.7]{PesZab07}.\n\n\\begin{remark}\nStricty speaking, in \\cite{MRT13} the integrands $f$ do not have to be predictable but only $\\cF_t\\otimes\\cB(U_1)$-adapted and $\\cF\\otimes\\cB([0,T])\\otimes\\cB(U_1)$-measurable. However, it is clear that in the case of predictable, i.e., $\\cP_T\\otimes\\cB(U_1)$-measurable, and square integrable Hilbert space-valued integrands $f$ the stochastic integral in \\cite{MRT13} coincides with the stochastic integral considered in \\cite{PesZab07}, \\cite{Pre10}. See \\cite{RueTap12} for a detailed comparison of the different spaces of integrands.\n\\end{remark}\n\nSince $\\bE\\int_0^T\\int_{U_1}\\nnrm{x}{U_1}^2\\,\\nu(\\ensuremath{\\mathrm{d}} x)\\ensuremath{\\mathrm{d}} t$ is finite for all $T<\\infty$, it follows that the integral process $(\\int_0^t\\int_{U_1} x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\geq 0}$ is uniquely determined (up to indistinguishability) as a $U_1$-valued square-integrable\nc\\`{a}dl\\`{a}g martingale. Taking into account the assumptions on the L\\'{e}vy process $L$, the L\\'{e}vy-Khinchin decomposition \\cite[Theorem~4.23]{PesZab07}, the definition of $q$, and the construction of the stochastic integral w.r.t.\\ $q$, it is not difficult to see that the processes $L$ and $(\\int_0^{t}\\int_{U_1} x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\geq0}$ are indistinguishable, i.e.,\n\\begin{equation}\\label{eq:L=qInt}\n\\bP\\Big(L(t)=\\int_0^t\\int_{U_1} x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\quad\\forall\\; t\\geq0\\Big)=1.\n\\end{equation}\n\nUsing Lemma~\\ref{lem:integrandIsomorphism}, we are now able to identify stochastic integrals w.r.t.\\ $L$ and stochastic integrals w.r.t.\\ the compensated Poisson random measure $q$. Recall from Remark~\\ref{not:Phix} that we identify processes $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ with the corresponding functions $\\kappa(\\Phi)\\in L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$. Thus, for such $\\Phi$ the integral process $(\\int_0^t\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ is defined.\n\n\\begin{lemma}\\label{lem:comparisonIntegrals}\nGiven $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$, the $H$-valued c\\`{a}dl\\`{a}g integral processes $(\\int_0^{t}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$ and $(\\int_0^{t\n}\\int_{U_1} \\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ are indistinguishable. That is,\n\\[\\bP\\Big(\\int_0^t\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)=\\int_0^t\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\quad \\forall\\,t\\in [0,T]\\Big)=1.\\]\n\\end{lemma}\n\n\\begin{proof}\nWe first assume that $\\Phi$ is a simple $\\ensuremath{\\mathscr L}(U_1,H)$-valued process of the form\n\\begin{equation*\n\\Phi(s)=\\sum_{k=0}^{m-1}\\mathds 1_{F_k}\\mathds 1_{(t_k,t_{k+1}]}(s)\\Phi_k,\\quad s\\in[0,T],\n\\end{equation*}\nwith $0\\leq t_0<t_1<\\cdots<t_m\\leq T$, $m\\in\\bN$, $F_k\\in\\cF_{t_k}$ and $\\Phi_k\\in\\ensuremath{\\mathscr L}(U_1,H)$. Recall from Section~\\ref{sec:LSEE} that $\\ensuremath{\\mathscr L}(U_1,H)$ is a subspace of $\\ensuremath{\\mathscr L_2}(U_0,H)$. Using \\eqref{eq:L=qInt} and applying standard arguments for the evaluation of stochastic integrals, we obtain for fixed $t\\in[0,T]$, $\\bP$-almost surely,\n\\begin{align*}\n\\int_0^t\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)&=\\sum_{k=0}^{m-1}\\mathds 1_{F_k}\\Phi_k(L(t_{k+1}\\wedge t)-L(t_k\\wedge t))\\\\\n&=\\sum_{k=0}^{m-1}\\mathds 1_{F_k}\\Phi_k\\Big(\\int_0^T\\int_{U_1} \\mathds 1_{(t_k\\wedge t, t_{k+1},\\wedge t]}(s) x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\Big)\\\\\n&=\\sum_{k=0}^{m-1}\\int_0^T\\int_{U_1} \\mathds 1_{F_k}\\mathds 1_{(t_k\\wedge t, t_{k+1},\\wedge t]}(s)\\Phi_k x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\\\\n&=\\int_0^t\\int_{U_1}\\Phi(s) x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x).\n\\end{align*}\nSince both processes are right-continuous, we see that the processes $(\\int_0^{t}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$ and $(\\int_0^{t\n}\\int_{U_1} \\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ are indistinguishable.\n\n\nFor general $\\Phi\\in L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$, take a sequence $(\\Phi_n)_{n\\in\\bN}$ of simple $\\ensuremath{\\mathscr L}(U_1,H)$-valued process\nsuch that $\\Phi_n\\to\\Phi$ in $L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$; see, e.g., \\cite[Proposition~2.3.8]{PreRoeck07} for a proof of the existence of such a sequence.\nThen, the\nprocesses $$\\int_0^{\\arg}\\Phi_n(s)\\,\\ensuremath{\\mathrm{d}} L(s)=(\\int_0^{t}\\Phi_n(s)\\,\\ensuremath{\\mathrm{d}} L(s))_{t\\in[0,T]}$$ and $\\int_0^{\\arg}\\int_{U_1}\\Phi_n(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)=(\\int_0^{t\n}\\int_{U_1} \\Phi_n(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x))_{t\\in[0,T]}$ are indistinguishable for all $n\\in\\bN$, and\nwe have the convergence $\\int_0^{\\arg}\\Phi_n(s)\\,\\ensuremath{\\mathrm{d}} L(s)\\to\\int_0^{\\arg}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)$ in $\\cM^2_T(H)$ by the construction of the stochastic integral w.r.t.\\ $L$.\n According to Lemma~\\ref{lem:integrandIsomorphism}, the convergence $\\Phi_n\\to\\Phi$ in $L^2(\\Omega_T,\\bP_T;\\ensuremath{\\mathscr L_2}(U_0,H))$ entails the convergengence $\\kappa(\\Phi_n)\\to \\kappa(\\Phi)$ in $L^2(\\Omega_T\\times U_1,\\bP_T\\otimes\\nu;H)$, so that we also have $$\\int_0^{\\arg}\\int_{U_1}\\Phi_n(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)\\to\\int_0^{\\arg}\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)$$ in $\\cM^2_T(H)$. Thus, $\\int_0^{\\arg}\\Phi(s)\\,\\ensuremath{\\mathrm{d}} L(s)=\\int_0^{\\arg}\\int_{U_1}\\Phi(s)x\\,q(\\ensuremath{\\mathrm{d}} s,\\ensuremath{\\mathrm{d}} x)$ as an equality in $\\cM^2_T(H)$, which yields the assertion.\n\\end{proof}\n\n\\end{appendix}\n\n\n\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we shall introduce a concept of {\\em fibred coarse embedding into Hilbert space}  for metric spaces (Definition 2.1) and prove the following result:\n\\par\n{\\bf Theorem 1.1.} {\\it Let $X$ be a discrete metric space with bounded geometry. If $X$ admits a fibred coarse embedding into Hilbert space, then the maximal coarse\nBaum-Connes conjecture holds for $X$.}\n\\par\nThe (maximal) coarse Baum-Connes conjecture \\cite{HR, Yu95, GWY} is a geometric analogue of the Baum-Connes conjecture \\cite{BC, BCH} and provides an algorithm  for computing the higher\nindices of generalized elliptic operators on non-compact spaces. It states\nthat a certain assembly map $\\mu$ or $\\mu_{\\max}$ from $\\lim_{d\\to \\infty} K_*(P_d(X))$ to $K_*(C^*(X))$ or $K_*(C^*_{\\max}(X))$ is an isomorphism, where $K_*(P_d(X))$ is the\nlocally finite $K$-homology group\nof the Rips complex $P_d(X)$,  and  $K_*(C^*(X))$ and $K_*(C^*_{\\max}(X))$ are respectively the $K$-theory groups of the Roe algebra and the maximal Roe algebra of $X$.\nThe conjecture has many applications in topology and geometry. In particular, it implies the Novikov conjecture on homotopy invariance of higher signatures,\nthe Gromov conjecture on non-existence of positive scalar curvature metrics on uniformly contractible Riemannian manifolds,\nand the zero-in-the-spectrum conjecture stating that the Laplacian operator acting on\nthe space of all $L^2$-forms of a uniformly contractible Riemannian manifold has zero in its spectrum.\n\\par\nRecall that a discrete metric space $X$ is said to have {\\em bounded geometry} if for any $r>0$ there is $N>0$ such that any ball of radius $r$ in $X$ contains at most $N$ elements.\nA map $f: X\\to Y$ from a metric space $X$ to another $Y$ is said to be a {\\em coarse embedding} or {\\em uniform embedding} \\cite{Grom93} if there exist non-decreasing functions\n$\\rho_1$ and $\\rho_2$ from $[0,\\infty)$ to $(-\\infty, \\infty)$ with $\\displaystyle\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1,2$) such that\n$$\\rho_1\\big( d(x,y) \\big) \\leq d\\big( f(x),f(y) \\big)\\leq \\rho_2\\big(d(x,y)\\big)$$\nfor all $x, y\\in X$.  \\quad M. Gromov suggested that coarse embeddability of a discrete group into a Hilbert space, or a certain Banach space, might be relevant to prove the Novikov conjecture\n\\cite{Grom93}. G. Yu subsequently proved the coarse Baum-Connes conjecture for discrete metric spaces with bounded geometry which are coarsely embeddable into Hilbert space \\cite{Yu00}.\nM. Gromov discovered that  a metric space of expander graphs \\cite{Marg, Lub} cannot be coarsely embedded into a Hilbert space \\cite{Grom00}. N. Higson showed that the assembly map $\\mu$ fails to be\nsurjective for certain Margulis-type of expanders \\cite{Hig99}.\nIn \\cite{Laf}, V. Lafforgue showed that there are residually finite groups whose\nassociated expander graphs cannot be coarsely embedded into any uniformly convex Banach space. Meanwhile, G. Gong, Q. Wang and G. Yu introduced the maximal Roe algebra\nof a metric space with bounded geometry  in \\cite{GWY}, and proved  a version of the maximal coarse Novikov conjecture, i.e. the injectivity of the maximal assembly map $\\mu_{\\max}$,\nfor the box space of a class of residually finite groups, including the expander graphs\nconstructed by V. Lafforgue.\nThereafter, the coarse Novikov conjecture, i.e. the injectivity of the assembly map $\\mu$,\n for a large class of expanders is proved in \\cite{CTWY, GTY, OY}.\nMore recently, H. Oyono-Oyono and G. Yu proved the maximal coarse Baum-Connes conjecture for certain metric spaces constructed from spaces with isometric actions by\nresidually finite groups \\cite{OY}, including a certain class of expander graphs. R. Willett and G. Yu proved the maximal coarse Baum-Connes conjecture for spaces of graphs with large\ngirth \\cite{WiY2}, also including a certain class of expander graphs.\n\\par\nThe notion of fibred coarse embedding into Hilbert space, which we are going to introduce,  generalizes Gromov's notion of coarse embedding into Hilbert space to a great extent.\nRoughly speaking, a metric space $X$ admits such an embedding implies that, although the whole space $X$ may not be coarsely embedded into Hilbert space globally, large bounded subsets of  $X$\ncan be coarsely embedded into Hilbert space within a common distortion as long as these large bounded subsets are far away towards infinity.\nThis feature provides the property with much flexibility, allowing many expander graphs studied above to admit such an embedding.\nIn particular, our result Theorem 1.1 includes the major results of \\cite{OY} and \\cite{WiY2} as special cases.\n\\par\nThis paper is organized as follows. In section 2, we introduce the notion of fibred coarse embedding into Hilbert space, and discuss sevaral situations to which this notion applies.\nIn section 3, we recall the definition of maximal Roe algebra and the formulation of the maximal coarse Baum-Connes conjecture. In section 4, we explain the strategy to prove Theorem 1.1.\nThe problem of proving the maximal coarse Baum-Connes conjecture for a metric space $X$ can be reduced to verifying that the evaluation homomorphism from the $K$-theory of a certain localization\nalgebra to the $K$-theory of the maximal Roe algebra at infinity for the coarse disjoint union of a sequence of finite subspaces of $X$ is an isomorphism.\nIn section 5, we define the twisted Roe algebra at infinity and its localization algebra for a sequence of finite metric spaces\nwhich admits a fibred coarse embedding into Hilbert space. In section 6, we discuss various ideals of the twisted algebras and show that the evaluation map for the twisted algebras\nis an isomorphism. In section 7, we prove a geometric analogue of the Bott periodicity in finite dimension by constructing the Bott maps and the Dirac maps as asymptotic morphisms. This is used to show\nthe evaluation map required in section 4 is an isomorphism, which implies Theorem 1.1.\n\\par\nThroughout this paper, we denote by $\\cK(H)$ or $\\cB(H)$ the algebras of all compact operators or all bounded linear operators on a Hilbert space $H$. We denote by\n$\\cK$ the algebra of all compact operators on a fixed separable Hilbert space $H_0$. Denote by $\\mathbb{N}=\\{0, 1, 2, \\cdots\\}$ the non-negative integers. Denote by $\\#A$ the number of elements in a set $A$.\nFor a metric space $X$, a point $x\\in X$ and $r>0$, we denote the ball of  $X$ with center $x$ and radius $r$  by $B(x, r)$,  or $B_X(x, r)$ if it is necessary to indicate the base space $X$.\n\n\\par\n{\\bf Acknowledgement.} The authors are very grateful to Rufus Willett who carefully read an early version of this paper and suggested very helpful comments.\n\n\n\n\\section{Fibred coarse embedding into Hilbert space}\nIn this section, we introduce the concept of {\\em fibred coarse embedding into Hilbert space} for a metric space, generalizing Gromov's notion of\n{\\em coarse embedding into Hilbert space} \\cite{Grom93}. Let $H$ be a separable Hilbert space, as a model space.\n\\par\n{\\bf Definition 2.1.} A metric space $(X, d)$ is said to admit a {\\em fibred coarse embedding into Hilbert space} if there exist\n\\begin{itemize}\n\\item a field of Hilbert spaces $(H_x)_{x\\in X}$ over $X$;\n\\item a section $s: X\\to \\bigsqcup_{x\\in X} H_x$ (i.e. $s(x)\\in H_x$);\n\\item two non-decreasing functions $\\rho_1$ and $\\rho_2$ from $[0, \\infty)$ to $(-\\infty, \\infty)$ with $\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1, 2$)\n\\end{itemize}\nsuch that for any $r>0$ there exists a bounded subset $K\\subset X$ for which there exists a ``trivialization''\n$$t_C: (H_x)_{x\\in C}\\longrightarrow C\\times H$$\nfor each subset $C\\subset X\\backslash K $ of diameter less than $r$, i.e. a map from $(H_x)_{x\\in C}$ to the constant field $C\\times H$ over $C$ such that the restriction  of $t_C$ to\nthe fiber $H_x$ ($x\\in C$) is an affine isometry $t_C(x): H_x\\to H$, satisfying\n\\begin{itemize}\n\\item[(1)] for any $x, y\\in C$, $\\rho_1(d(x, y))\\leq \\|t_C(x)(s(x))-t_C(y)(s(y))\\|\\leq \\rho_2(d(x, y))$;\n\\item[(2)] for any two subsets $C_1, C_2\\subset X\\backslash K$ of diameter less than $r$ with $C_1\\cap C_2\\not=\\emptyset$, there exists an affine isometry $t_{C_1C_2}: H\\to H$ such that\n\t\t$t_{C_1}(x)\\circ t_{C_2}^{-1}(x)=t_{C_1C_2}$ for all $x\\in C_1\\cap C_2$.\n\\end{itemize}\n\\hfill{$\\Box$}\n\\par\nIn the following we discuss several situations to which the above notion applies. Let $(X_n)_\\nn$ be a sequence of bounded metric spaces.\n\\par\nRecall that {\\em a coarse disjoint union of $(X_n)_\\nn$} is the disjoint union $X=\\bigsqcup_\\nn X_n$ equipped with a metric $d$ such that (1) the restriction of $d$ to each $X_n$ is the original metric\nof $X_n$; (2) $d(X_n, X_m)\\to \\infty$ as $n+m\\to \\infty$ and $n\\neq m$. Note that any two metrics satisfying these two conditions are coarsely equivalent. So we may refer to $X$ as {\\em the} coarse disjoint\nunion of  $(X_n)_\\nn$.\n\\par\nFor each $\\nn$, a metric space $\\wt{X_n}$ is called {\\em a Galois covering of $X_n$} if there exists a discrete group $\\Gamma_n$ acting on $\\wt{X_n}$ freely and properly by isometries\nsuch that $X_n=\\wt{X_n}\/\\Gamma_n$. Denote by $\\pi_n: \\wt{X_n}\\to X_n$ the associated covering map.\n\\par\nA sequence of Galois coverings $(\\wt{X_n})_\\nn$ of $(X_n)_\\nn$ is said to be {\\em asymptotically faithful} (cf. \\cite{WiY1}) if for any $r>0$ there exists $N\\in \\mathbb{N}$ such that, for all\n$n\\geq N$, the covering map $\\pi_n: \\wt{X_n}\\to X_n$ is ``$r$-isometric'', i.e. for all subsets $\\wt{C}\\subset \\wt{X_n}$ of diameter less than $r$, the restriction of $\\pi_n$ to $\\wt{C}$ is an isometry\nonto $C=\\pi_n(\\wt{C})\\subset X_n$.\n\\par\nA sequence of Galois coverings $(\\wt{X_n})_\\nn$ of $(X_n)_\\nn$ is said to admit a {\\em uniform equivariant coarse embedding into Hilbert space} if there exist a map $f_n: \\wt{X_n}\\to H$ for each\n$\\nn$, and two non-decreasing functions $\\rho_1$ and $\\rho_2$ from $[0, \\infty)$ to $(-\\infty, \\infty)$ with $\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1, 2$), such that\n(1) $f_n$ is $\\Gamma_n$-equivariant (this implies $H$ has an action of $\\Gamma_n$ by affine isometries) for all $\\nn$; (2) $\\rho_1(d(x, y))\\leq \\|f_n(x)-f_n(y)\\|\\leq \\rho_2(d(x, y))$\nfor all $x, y\\in \\wt{X_n}$, $\\nn$.\n\\par\n{\\bf Theorem 2.2.} {\\it\nLet $X=\\bigsqcup_\\nn X_n$ be a coarse disjoint union of a sequence of bounded metric spaces. If there exists a sequence of asymptotically faithful Galois coverings\n$(\\wt{X_n})_\\nn$ which admits a uniform equivariant coarse embedding into Hilbert space, then $X$ admits a fibred coarse embedding into Hilbert space.\n}\n\\par\n{\\bf Proof.} For each $\\nn$,  the group $\\Gamma_n$ acts on $\\wt{X_n}\\times H$ by\n$$\\gamma \\cdot (x, h)= (\\gamma x, \\gamma h)$$\nfor $x\\in \\wt{X_n}, h\\in H, \\gamma\\in \\Gamma_n$. Denote the orbit of $(x, h)\\in \\wt{X_n}\\times H$ by $[(x, h)]$, i.e.\n$$[(x, h)]=\\{ \\, (\\gamma x, \\gamma h) \\; | \\; \\gamma\\in \\Gamma_n \\, \\}.$$\nFor any point $a\\in X_n=\\wt{X_n}\/\\Gamma_n$, the action\nof a group element $\\gamma\\in \\Gamma_n$ on $\\wt{X_n}$ permutes the points in $\\pi_n^{-1}(a)\\subset \\wt{X_n}$. Define\n$$H_a:=\\Big( \\pi_n^{-1}(a) \\times H \\Big)\\Big\/ \\Gamma_n.$$\nThen\n$$ (H_a)_{a\\in X_n}=\\Big( \\wt{X_n} \\times H \\Big)\\Big\/\\Gamma_n $$\nis a field of Hilbert spaces over $X_n$. Define a section\n$$s: X_n\\longrightarrow \\Big( \\wt{X_n} \\times H \\Big)\\Big\/\\Gamma_n$$\nby the formula\n$$s(a)=[(x, f_n(x))]\\in H_a$$\nfor any $x\\in \\pi_n^{-1}(a)$, $a\\in X_n$. This is well-defined since the map $f_n: \\wt{X_n}\\to H$ is $\\Gamma_n$-equivariant.\n\\par\nSince the covering sequence $(\\wt{X_n})_\\nn$ is asymptotically faithful, for any $r>0$ there exists $N\\in \\mathbb{N}$ such that\nfor any $n\\geq N$ and any $C\\subset X_n$ of diameter less than $r$,\nthe action of an element $\\gamma \\in \\Gamma_n$ on $\\wt{X_n}$ permutes the (disjoint) components of $\\pi_n^{-1}(C)$, and the restriction of $\\pi_n$ to each of these components is an\nisometry onto $C$. For a point $z\\in \\pi_n^{-1}(C)$, denote by ${\\wt{C}}^z$ the component of $\\pi_n^{-1}(C)$ containing $z$. Then each such\ncomponent ${\\wt{C}}^z$ gives rise to a trivialization:\n$$t_{C, z}: (H_a)_{a\\in C}=\\Big( \\pi_n^{-1}(C) \\times H \\Big)\\Big\/ \\Gamma_n  \\longrightarrow C\\times H, $$\nwhere, for any $a\\in C\\subset X_n$, the affine isometry\n$$t_{C, z}(a): \\; \\; H_a=\\Big( \\pi_n^{-1}(a) \\times H \\Big)\\Big\/ \\Gamma_n     \\longrightarrow H$$\nis given by\n$$\\Big( t_{C, z}(a) \\Big) \\Big( [(x, h)] \\Big) =h$$\nfor $[(x, h)]\\in H_a$ represented by $(x, h)\\in \\wt{C}^z\\times H$.\n\\par\nNow, for any $a, b\\in C$, there exist $x, y\\in \\wt{C}^z$ such that $\\pi_n(x)=a$ and $\\pi_n(y)=b$, so that\n\\[\n\\begin{array}{c}\n  \\Big\\|\\Big(t_{C, z}(a)\\Big) \\big( s(a) \\big) - \\Big( t_{C, z}(b) \\Big) \\big( s(b) \\big) \\Big\\| = \\Big\\| f_n(x)-f_n(y) \\Big\\|              \\\\\n\\in  \\big[ \\rho_1(d(x, y)), \\; \\rho_2(d(x, y)) \\big] =  \\big[ \\rho_1(d(a, b)), \\; \\rho_2(d(a, b)) \\big].\n\\end{array}\n\\]\n\\par\nMoreover, for any $C_1, C_2\\subset X_n$ ($n\\geq N$) of diameter less than $r$ with $C_1\\cap C_2\\not=\\emptyset$, there exist $z, \\gamma z\\in \\wt{X_n}$ for some $\\gamma\\in \\Gamma_n$ such that\n$\\pi_n(z)=\\pi_n(\\gamma z)\\in C_1\\cap C_2$. Then,  for all $a\\in C_1\\cap C_2$,\n$$t_{C_1, z}(a)\\circ t_{C_2, \\gamma z}^{-1}(a)=\\gamma: \\; H\\to H$$\nas an affine isometry mapping $h\\in H$ to $\\gamma h\\in H$. The proof is complete.\n\\hfill{$\\Box$}\n\\par\n{\\bf Example 2.3.} If a  metric space $X$ is coarsely embeddable into Hilbert space, then clearly it admits a fibred coarse embedding into Hilbert space. The maximal coarse Baum-Connes conjecture in this\ncase follows from G. Yu's  work on the coarse Baum-Connes conjecture \\cite{Yu00}, together with a recent result by J. \\v{S}pakula and R. Willett that $K_*(C^*_{\\max}(X))$ is isomorphic to $K_*(C^*(X))$ if $X$ is coarsely embeddable into Hilbert space \\cite{SpWi}.  Note also that for a discrete group, fibred coarse embeddability is the same as usual coarse embeddability.\n\\hfill{$\\Box$}\n\\par\n{\\bf Example 2.4.} Let $X$ be a discrete metric space with bounded geometry. Suppose a residually finite group $\\Gamma$ acts on $X$ freely, properly and cocompactly by isometries,\n such that $X$ is $\\Gamma$-equivariantly coarsely embeddable into Hilbert space. Let $(\\Gamma_n)_\\nn$ be a sequence of finite index normal subgroups of $\\Gamma$ such that for all $r>0$ there\nexists $N\\in \\mathbb{N}$ so that if $B(e, r)$ is the ball in $\\Gamma$ of radius $r$ about the identity, then $\\Gamma_n\\cap B(e, r)=\\{e\\}$ for all $n\\geq N$.\nLet $X_n=X\/\\Gamma_n$ be the quotient space. It follows from Theorem 2.2 that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding into Hilbert space. Indeed, the constant sequence\n$\\wt{X_n}=X$ ($\\nn$)\nserves as the asymptotically faithful Galois coverings.  Theorem 1.1 implies the maximal coarse Baum-Connes conjecture for such X.\nA special case of this result was proved by H. Oyono-Oyono and G. Yu in \\cite{OY}.\n\\par\nFor a residually finite group $\\Gamma$, the space $X(\\Gamma)=\\bigsqcup_{\\nn} \\Gamma\/\\Gamma_n$ is called {\\em the box space of  $\\Gamma$} (\\cite{Roe03}). It turns out that if the box space $X(\\Gamma)$ is coarsely embeddable into Hilbert space, then $\\Gamma$ is a-T-menable\n(\\cite{Roe03}). It is also easy to see that the converse of this implication is not true. However, it is shown in \\cite{CWW12} that $\\Gamma$ is a-T-menable if and only if the box space $X(\\Gamma)$ is fibred coarsely embeddable into Hilbert space.\n\\hfill{$\\Box$}\n\\par\n{\\bf Example 2.5.} Recall that the {\\em girth} of a graph $G$, denoted by $girth(G)$,  is the shortest length of a cycle in $G$. A sequence of finite connected graphs $(G_n)_\\nn$ is\nsaid to have {\\em large girth} if $girth(G_n)\\to \\infty$ as $n\\to \\infty$. Then the coarse disjoint union $X=\\bigsqcup_{\\nn} G_n$ admits a fibred coarse embedding into Hilbert space.\nIndeed, let $\\wt{G_n}$ be the {\\em universal cover of $G_n$}, which is actually a tree. Then the covering sequence $(\\wt{G_n})_\\nn$ satisfies the conditions in Theorem 2.2. The maximal coarse Baum-Connes\nconjecture for such $X$ was proved by R. Willett and G. Yu in \\cite{WiY2}.\n\\hfill{$\\Box$}\n\\par\nNote that Examples 2.4 and 2.5 imply that a large class of expander graphs admit a fibred coarse embedding into Hilbert space.\n\n\n\n\n\\section{The maximal coarse Baum-Connes conjecture}\n\n\nIn this section, we shall collect from \\cite{GWY} results concerning the maximal Roe algebra of a proper metric space with bounded geometry and the formulation of the maximal coarse Baum-Connes conjecture\n(see also \\cite{OY, WiY1, WiY2}).\n\\par\nLet $X$ be a proper metric space (a metric space is called $proper$ if every closed ball is compact). An $X$-module $H_X$ is a separable Hilbert space\nequipped with a  $*$-representation $\\pi$ of $C_0(X)$, the algebra of all continuous functions on $X$ which vanish\nat infinity. An $X$-module is called non-degenerate if the $*$-representation of $C_0(X)$ is non-degenerate.\nAn $X$-module is said to be standard if no nonzero function in $C_0(X)$ acts as a compact operator.\nWhen $H_X$ is an $X$-module,  for each $f\\in C_0(X)$ and $h\\in H_X$, we denote $(\\pi (f))h$ by $fh$.\n\n\\par\n{\\bf Definition 3.1.} (cf. \\cite{Roe93})  Let $X$ be a standard non-degenerate $X$-module.\n\\\\ \\indent (1) The {\\em support} $\\supp(T)$ of a bounded linear operator $T: H_X\\to H_X$ is defined to be the complement of the set of all points $(x, y)\\in X\\times X$ for which\nthere exist $f, g\\in C_0(X)$  such that $gTf=0$ but $f(x)\\not= 0$, $g(y)\\not=0$.\n\\\\ \\indent (2) A bounded operator $T: H_X\\to H_X$  is said to have {\\em finite propagation}  if\n$$\\sup\\{d(x, y): (x, y)\\in \\Supp(T)\\}<\\infty.$$\nThis number is called the {\\em propagation of $T$}.\n\\\\ \\indent (3) A bounded operator $T: H_X\\to H_X$ is said to be {\\em locally compact} if the operators $fT$ and $Tf$ are compact for all $f\\in C_0(X)$.\n\\par\nDenote by $\\mathbb{C}[X, H_X]$, or simply $\\mathbb{C}[X]$, the set of all locally compact, finite propagation operators on a standard non-degenerate $X$-module $H_X$.  It is straightforward to check that $\\mathbb{C}[X]$ is a $*$-algebra which,\nup to non-canonical isomorphisms, does not depend on the choice of standard non-degenerate $X$-module.\n\\par\n{\\bf Definition 3.2.} A {\\em net} of a metric space $X$ is a countable subset $\\Gamma\\subset X$ such that there exist numbers $\\delta>0$, $R>0$ satisfying (1) $d(\\gamma, \\gamma')>\\delta$\nfor all distinct elements $\\gamma, \\gamma'\\in \\Gamma$; (2) for any $x\\in X$ there exists $\\gamma\\in \\Gamma$ such that $d(x, \\gamma)<R$.  \\quad A metric space $X$ is said to have {\\em bounded geometry}\nif $X$ contains a net with bounded geometry.\n\\par\nThe following result is essentially proved in \\cite{GWY}.\n\\par\n{\\bf Lemma 3.3.} {\\it Let $X$ be a proper metric space with bounded geometry, and let $H_X$ be a standard non-degenerate $X$-module. For any $r>0$ there exists a constant $c>0$ such that\nfor any $*$-representation $\\phi$ of $\\cx$ on a Hilbert space $H_\\phi$ and any $T\\in \\cx$ with propagation less than $r$, we have\n$$||\\phi(T)||_{\\cB(H_\\phi)}\\leq c  \\, || T ||_{\\cB(H_X)} \\,. $$\n}\n\\par\nThis allows us to define the maximal Roe algebra of $X$.\n\\par\n{\\bf Definition 3.4.} \\quad (\\cite{GWY}) Let $X$ be a proper metric space with bounded geometry. The {\\em maximal Roe algebra} of $X$, denoted by $\\cmx$, is the completion of\n$\\cx$ with respect to the $C^*$-norm:\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cx\\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\par\n{\\bf Definition 3.5.}\nLet $H_X=\\ell^2(Z, H_0)$, where $Z\\subset X$ is a countable dense subset of $X$ and $H_0$ is an infinite dimensional separable Hilbert space. A function $f\\in C_0(X)$ acts on\n$\\ell^2(Z)\\otimes H_0$ by pointwise multiplication on the first component: $f(\\xi\\otimes h)=f\\xi\\otimes h$. Define $\\mathbb{C}_f [X]$ to be the $*$-algebra of all bounded functions\n$T: Z\\times Z\\to \\cK:=\\cK(H_0)$, also viewed as $Z\\times Z$-matrices, such that\n\\begin{itemize}\n\\item for any bounded subset $B\\subset X$, the set\n\t\t$$\\{(x, y)\\in B\\times B\\cap Z\\times Z \\; | \\; T(x, y)\\not= 0 \\}$$\n\t\t is finite;\n\\item there exists $L>0$ such that\n\t\t$$\\#\\{y\\in Z \\; | \\; T(x, y)\\not= 0 \\}<L,  \\quad \\#\\{y\\in Z \\; | \\; T(y, x)\\not= 0 \\}<L $$\n\t\tfor all $x\\in Z$;\n\\item there exists $R>0$ such that $T(x, y)=0$ whenever $d(x, y)>R$ for $x, y\\in Z$.\n\\end{itemize}\nNote that in general $\\mathbb{C}_f [X]$ is a dense $*$-subalgebra of $\\mathbb{C}[X, \\ell(Z, H_0)]$ within $C^*_{\\max}(X)$. In the sequel, we will use $\\mathbb{C}_f [X]$ to replace $\\mathbb{C}[X]$ as a generating subalgebra of $C^*_{\\max}(X)$.\n\\par\n{\\bf Remark 3.6.} (cf. \\cite{HRY} and \\cite{GWY}) \\quad (1) $\\cmx$ does not depend on the choice of countable dense subset $Z\\subset X$ up to non-canonical isomorphisms.\nIf $X$ and $Y$ are coarsely equivalent, then $\\cmx$ is isomorphic to $\\cmy$ via a non-canonical isomorphism. (2) The $K$-theory groups of $\\cmx$ do not depend on the choice of $Z$ up to\ncanonical isomorphisms. If $X$ and $Y$  are coarsely equivalent, then $K_*(\\cmx)$ is isomorphic to $K_*(\\cmy)$ via a canonical isomorphism.\n\\par\nWe next define the assembly map $\\mu_{\\max}$ (also referred to as the index map or the Baum-Connes map) for the maximal Roe algebras.\n\\par\nLet $X$ be a proper metric space. Recall that the locally finite $K$-homology groups $K_i(X)=KK_i(C_0(X),\\mathbb{C})$ $(i=0,1)$ are generated by certain cycles\n(abstract elliptic operators) modulo  certain equivalence relations \\cite{Kas75, Kas88}:\n\\\\ \\indent (1) a cycle for $K_0(X)$ is a pair $(H_X, F)$, where $H_X$ is an $X$-module and $F$ is a bounded linear operator acting on $H_X$ such that $F^*F-I$ and $FF^*-I$ are locally compact,\nand $\\phi F-F\\phi$ is compact for all $\\phi\\in C_0(X)$;\n\\\\ \\indent (2) a cycle for $K_1(X)$ is a pair $(H_X, F)$, where $H_X$ is an $X$-module and $F$ is a {\\em self-adjoint} operator acting on $H_X$ such that $F^2-I$ is locally compact,\nand $\\phi F-F\\phi$ is compact for all $\\phi\\in C_0(X)$.\n\\par\nIn both cases, the equivalence relations on cycles are given by homotopy of the operators $F$, unitary equivalence, and direct sum with ``degenerate\" cycles, those cycles for which\n$F\\phi-\\phi F$, $\\phi(F^* F-I)$ and so on, are not merely compact but actually zero \\cite{Kas75, Kas88}.\n\\par\nThe assembly map $\\mu_{\\max}: K_*(X) \\to K_*(\\cmx)$ is defined as follows.\n\\par\n{\\bf Definition 3.7.} Let $(H_X, T)$ represent a cycle in $K_0(X)$. For any $R>0$, one can always take a  locally finite,\nuniformly bounded open cover $\\{U_i\\}_i$ of $X$ such that\n$$diameter(U_i)<R $$\nfor all $i$, and a continuous partition of unity  $\\{\\phi_i\\}_i$ subordinate to the open cover $\\{U_i\\}_i$. Define $F=\\sum_i \\phi_i^{\\frac{1}{2}}T\\phi_i^{\\frac{1}{2}}$, where the sum converges in\nthe strong operator topology. It is not hard to see that $(H_X, T)$ and $(H_X, F)$ are equivalent in $K_0(X)$. Note that now the propagation of $F$ is less than or equal to $R$, so that\n$F^*F-I$ and $FF^*-I$ are in $\\mathbb{C}[X]$. Let (cf. \\cite{Mil})\n\\[W=\n\\left(\n\\begin{array}{cc}\nI & F \\\\\n0 & I\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n-F^* & I\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\nI & F \\\\\n0 & I\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\n0 & -I \\\\\nI & 0\n\\end{array}\n\\right)\n\\in \\cB(H_X\\oplus H_X).\n\\]\nThen both $W$ and $W^{-1}$ have finite propagation (at most $3R$), and\n\\[\nW\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\nW^{-1}-\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\n\\in \\mathbb{C}[X]\\otimes \\cM_2(\\mathbb{C}).\n\\]\nWe then define\n\\[\n\\mu \\big( [(H_X, T)] \\big):=\n\\left[\nW\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\nW^{-1}\n\\right]\n-\n\\left[\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\n\\right]\n\\]\nin $K_0(\\mathbb{C}[X])$. Furthermore, $\\mu \\big( [(H_X, T)] \\big)$ defines an element in $K_0(C^*_{\\max}(X))$ by considering $\\mathbb{C}[X]$ as a  $*$-subalgebra of $C^*_{\\max}(X)$. This element is denoted by $\\mu_{\\max} \\big( [(H_X, T)] \\big) \\in K_0(C^*_{\\max}(X))$.\nThus, we obtain the assembly map $\\mu_{\\max}: K_0(X)\\rightarrow K_0(C^*_{\\max}(X))$. Similarly, we can define $\\mu_{\\max}: K_1(X)\\rightarrow K_1(C^*_{\\max}(X))$.\n\\par\n{\\bf Remark 3.8.}\nLet $X$ be a proper metric space with bounded geometry as above, and $Z$ a countable dense subset of $X$ used to define $\\mathbb{C}_f [X]$ as in Definition 3.5. For any natural number $n>0$, let $Z_n$ be a subset of $Z$ such that $d(x, y)>\\frac{1}{2n}$ for distinct $x, y\\in Z_n$, and $d(x, Z_n)\\leq \\frac{1}{n}$ for all $x\\in X$. Without loss of generality, we may assume $Z=\\cup_{n=1}^\\infty Z_n$. Let $\\mathbb{C}[Z_n]$ be the $*$-algebra of all bounded functions $T: Z_n\\times Z_n\\to \\mathcal{K}$ with finite propagation, i.e., there exists $R>0$ such that $T(x, y)=0$ whenever $d(x, y)>R$ for all $x, y\\in Z_n$. Then $\\mathbb{C}[Z_n]\\subset \\mathbb{C}_f [X]$. Moreover, there exists a non-canonical $*$-isomorphism (cf. \\cite{HRY} or 4.4 in \\cite{GWY}) $Ad(U): \\mathbb{C}[X]\\to \\mathbb{C}[Z_n]$ such that, if $T\\in \\mathbb{C}[X]$ has propagation less than $R$, then the propagation of $(Ad(U))(T)$ is less than $R+\\frac{2}{n}$.\n\\par\nFor any $R>0$, let $\\mu \\big( [(H_X, T)] \\big)\\in K_0(\\mathbb{C}[X])$ be as in Definition 3.7. Then\n$$Ad(U)_*\\big(\\mu \\big( [(H_X, T)] \\big) \\big)\\in K_0(\\mathbb{C}[Z_n]),$$\nwhich also defines an element in $K_0(\\mathbb{C}_f[X])$ via the inclusion $\\mathbb{C}[Z_n]\\hookrightarrow \\mathbb{C}_f [X]$.\nNote that the propagation of the element\n\\[\n\\big( Ad(U) \\big)\\left(\nW\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\nW^{-1}-\n\\left(\n\\begin{array}{cc}\nI & 0 \\\\\n0 & 0\n\\end{array}\n\\right)\\right)\n\\in \\mathbb{C}_f[X]\\otimes \\cM_2(\\mathbb{C})\n\\]\nis less than $6R+\\frac{2}{n}$. Note also that $Ad(U)_*$ canonically induces the identity on $K_*(C^*_{\\max}(X))$.\n\\par\n{\\bf Definition 3.9.} Let $X$ be a discrete metric space with bounded geometry. For each $d\\geq 0$, the {\\em Rips complex} $P_d(X)$ at scale $d$ is defined to be the simplicial polyhedron\nin which the set of  vertices is $X$, and a finite subset $\\{x_0, x_1, \\cdots, x_q\\}\\subset X$ spans a simplex if and only if $d(x_i,x_j)\\leq d$ for all $0\\leq i, j\\leq q$.\n\\par\nEndow $P_d(X)$ with the {\\em spherical metric}. Recall that on each path connected component of $P_d(X)$, the spherical\nmetric is the maximal metric whose restriction to each simplex\n$\\{\\sum_{i=0}^q t_ix_i| t_i\\geq 0, \\sum_{i=0}^q t_i=1\\}$ is the metric obtained by identifying the simplex with\n$S_{+}^q$ via the map\n$$\\sum_{i=0}^q t_ix_i \\mapsto \\left( \\frac{t_0}{\\sqrt{\\sum_{i=0}^q t_i^2}}, \\frac{t_1}{\\sqrt{\\sum_{i=0}^q t_i^2}}, \\cdots,\n\\frac{t_q}{\\sqrt{\\sum_{i=0}^q t_i^2}} \\right)\n$$\nwhere $S_+^q:=\\{(s_0, s_1, \\cdots, s_q)\\in \\mathbb{R}^{q+1}, s_i\\geq 0, \\sum_{i=0}^q s_i =1\\}$ is endowed with the standard\nRiemannian metric. If $y_0, y_1$ belong to two different connected components $Y_0, Y_1$ of $P_d(X)$, we define\n$$d(y_0, y_1)= \\min \\{d(y_0, x_0)+d_X(x_0, x_1)+d(x_1, y_1)| x_0\\in X\\cap Y_0, x_1\\in X\\cap Y_1\\}.$$\nThe topology induced by the above metric is the same as the weak topology of the simplicial complex:\na subset $S\\subset P_d(X)$ is closed if and only if the intersection of $S$ with each simplex is closed.\n\\par\nNote that for any $d\\geq 0$, $\\pdx$ is coarsely equivalent to $X$ via the inclusion map. If $d<d'$, then $P_d(X)$ is included in $P_{d'}(X)$ as a subcomplex via a simplicial map.\nPassing to inductive limit, we obtain the assembly map\n$$ \\mu_{\\max}: \\lim_{d\\to \\infty} K_*(\\pdx) \\to  \\lim_{d\\to \\infty} K_*(\\cmpdx) \\cong K_*(\\cmx) .$$\n\\par\n{\\bf The maximal coarse Baum-Connes conjecture:}  {\\it\nIf $X$ is a discrete metric space with bounded geometry, then the assembly map $\\mu_{\\max}$ is an isomorphism.\n}\n\\par\n{\\bf Remark 3.10.} The maximal coarse Baum-Connes conjecture provides a method to compute the higher indices of elliptic differential operators\n(Dirac operators) on non-compact complete Riemannian manifolds and has many applications in topology and geometry. In particular, it implies the Novikov conjecture, the Gromov positive scalar curvature conjecture,\netc. (cf. e.g. \\cite{HR, Roe93, Roe96, Yu95, Yu98, Yu00, Yu06, GWY}).\n\n\n\n\n\n\n\n\n\n\\section{Reduction to coarse disjoint union of finite metric spaces}\n\nThe strategy to prove Theorem 1.1 is as follows. Let $X$ be a discrete metric space with bounded geometry which admits a fibred coarse embedding into Hilbert space.\nWe partition $X$ as $X=X^{(0)}\\cup X^{(1)}$ such that each of $X^{(0)}$, $X^{(1)}$ and $X^{(0)}\\cap X^{(1)}$ is a coarse disjoint union of a sequence of finite subspaces\nof $X$, and the excision pair $(X^{(0)}, \\; X^{(1)})$ respects the Mayer-Vietoris exact sequences on both the $K$-homology for these subspaces and the $K$-theory for the\ncorresponding maximal Roe algebras (the method of cutting the whole space into coarse disjoint unions of finite subspaces has its origin in \\cite{WiY-book}). Consequently, it suffices to prove Theorem 1.1 for $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ being a coarse disjoint union of a sequence of finite\nmetric spaces. This case in turn can be reduced to verifying the evaluation homomorphism from the $K$-theory groups of certain localization algebras to that of the corresponding\n{\\em maximal Roe algebra at infinity for the sequence $(X_n)_\\nn$} to be an isomorphism, a fact which will be proved in sections 5, 6 and 7. \n\\par\nIn this section,  we shall explain in detail the above process of reduction. We start with the story for sequences of finite metric spaces. Let $(X_n)_\\nn$ be a sequence of finite (discrete) metric spaces with {\\em uniform bounded geometry},\ni.e. for any $r>0$ there exists $N>0$ such that $\\# B(x, r)<N$ for all $x\\in X_n, \\nn$. For each $d\\geq 0$, let $P_d(X_n)$ be the Rips complex of $X_n$ at scale $d$ endowed with\nthe spherical metric. Take a countable dense subset $\\zdn \\subset \\pdxn$ for each $d\\geq 0$ in such a way that $\\zdn\\subseteq Z_{d', n}$ whenever $d<d'$, for all $\\nn$.\n\\par\n{\\bf Definition 4.1.} For each $d\\geq 0$, define $\\cupdxn$ to be the set of all equivalence classes $\\ttt$ of sequences $(T^{(0)}, \\cdots, T^{(n)}, \\cdots)$ described as follows:\n\\begin{itemize}\n\\item[(1)] $T^{(n)}$ is a bounded function from $\\zdn \\times \\zdn$ to $\\cK$ for all $\\nn$;\n\\item[(2)] for any bounded subset $B\\subset \\pdxn$, the set\n\t\t\t$$\\{ (x, y)\\in B\\times B \\cap \\zdn\\times \\zdn \\; | \\; T^{(n)} (x, y)\\not= 0 \\}$$\n\t\t\tis finite;\n\\item[(3)] there exists $L>0$ such that\n\t\t\t$$\\#\\{y\\in \\zdn \\; | \\; T^{(n)}(x, y)\\not= 0 \\}<L,  \\quad  \\#\\{y\\in \\zdn \\; | \\; T^{(n)}(y, x)\\not= 0 \\}<L$$\n\t\t\tfor all $x\\in \\zdn, \\nn$;\n\\item[(4)] there exists $R>0$ such that $T^{(n)}(x, y)=0$ whenever $d(x, y)>R$ for $x, y\\in \\zdn, \\; \\nn$.\n\\end{itemize}\nThe equivalence relation $\\sim$ on these sequences is defined by\n$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots) \\sim (S^{(0)}, \\cdots, S^{(n)}, \\cdots)$$\n if and only if\n$$\\lim_{n\\to \\infty} \\sup_{x, y\\in \\zdn} \\big\\| T^{(n)}(x, y) - S^{(n)}(x, y) \\big\\|_\\cK =0.$$\nViewing $T^{(n)}$ as $\\zdn\\times \\zdn$ matrices, $\\cupdxn$ is then made into a $*$-algebra by using the usual matrix operations.\n\\par\nDefine $\\cumpdxn$ to be the completion of $\\cupdxn$ with respect to the norm\n\\begin{footnotesize}\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cupdxn \\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\end{footnotesize}\n\\hfill{$\\Box$}\n\\par\nNote that $\\|T\\|_{\\max}$ is well-defined since $(X_n)_\\nn$ have uniform bounded geometry. Moreover, $(\\pdxn)_\\nn$ is uniformly coarsely equivalent to $(X_n)_\\nn$ for\nany $d>0$, so that\n$\\cumpdxn$ is isomorphic to $C^*_{u, \\max, \\infty}((X_n)_\\nn)$ via a non-canonical isomorphism. Recall from Definition 3.7 and Remark 3.8 that the individual assembly maps\n$$\\mu_{\\max}: K_*(\\pdxn) \\to K_*(\\cmpdxn)$$\n($\\nn$) can be defined by $\\mu_{\\max} ([(H_X, T)])=\\mu_{\\max} ([(H_X, F)])$ such that the propagation of $F$ is less than any given small $R>0$ which is independent of $\\nn$, and that $\\|F\\|\\leq 1$. Consequently,\nwe can obtain the following assembly map at infinity:\n$$\\mu_{\\max, \\infty}: \\; \\; \\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)} \\longrightarrow  K_*\\Big(  \\cumpdxn  \\Big).$$\n\\par\nThe following notion of localization algebra has its origin in \\cite{Yu97}.\n\\par\n{\\bf Definition 4.2.}  For each $d\\geq 0$, define $\\culpdxn$  to be the $*$-algebra of all bounded and uniformly norm-continuous functions\n$$f: \\; [0,\\infty)\\longrightarrow \\cupdxn$$\nsuch that $f(t)$ is of the form $f(t)=[(f^{(0)}(t), \\cdots, f^{(n)}(t), \\cdots )]$ for all $t\\in [0, \\infty)$, where the family of functions\n$(f^{(n)}(t))_{\\nn, t\\geq 0}$ satisfy the conditions in Definition 4.1 with uniform constants, and there exists a bounded function $R:\\; [0, \\infty)\\to [0,\\infty)$ with\n$\\lim_{t\\to \\infty} R(t)=0$ such that\n$$ \\big( f^{(n)}(t)\\big) (x, y) =0  \\quad \\mbox{whenever} \\quad d(x, y)>R(t)$$\nfor all $x, y\\in \\zdn$, $\\nn$, $t\\in [0, \\infty)$.\n\\par\nDefine $\\culmpdxn$ to be the completion of $\\culpdxn$ with respect to the norm\n$$\\big\\| f \\big\\|_{\\max} :=\\sup_{t\\in [0, \\infty)} \\big\\|f(t)\\big\\|_{\\max}.$$\n\\hfill{$\\Box$}\n\\par\nWe can also define a local assembly map at infinity as in \\cite{Yu97}:\n$$\\mu_{L, \\max,\\infty}: \\; \\; \\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)} \\longrightarrow  K_*\\Big(  \\culmpdxn  \\Big).$$\n\\par\n{\\bf Proposition 4.3.}  {\\it\nSuppose $(X_n)_\\nn$ have uniform bounded geometry. Then the local assembly map $\\mu_{L, \\max,\\infty}$ is an isomorphism.\n}\n\\hfill{$\\Box$}\n\\par\nThe proof is similar to the arguments in \\cite{Yu97}. Note that the evaluation homomorphism\n$$e: \\; \\culmpdxn \\longrightarrow \\cumpdxn $$\ndefined by $e(f)=f(0)$ induces the following commutative diagram:\n\n\\[\n\\xymatrix{\n & \\;\\;\\;\\;\\;\\; \\displaystyle    \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxn  \\big) \\ar[d]^{e_*}            \\\\\n\\displaystyle\\lim_{d\\to\\infty}     \\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)}    \\;\\; \\;\\;\\;  \\ar[ur]^{\\mu_{L, \\max, \\infty}}_\\cong  \\ar[r]^{\\mu_{\\max, \\infty}\\quad\\quad\\quad}\n&\n\\;\\;\\;\\;\\;\\;\\;    \\displaystyle   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxn  \\big).\n  }\n\\]\n\n\\par\nWe will devote the second half of the paper, section 5, 6 and 7, to prove the following result.\n\\par\n{\\bf Theorem 4.4.} {\\it\nLet $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry. If the coarse disjoint union $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ admits a fibred coarse embedding into\nHilbert space, then\n\\begin{small}\n$$e_*:\\; \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxn  \\big) \\longrightarrow   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxn  \\big)$$\n\\end{small}\nis an isomorphism. Consequently, $\\mu_{\\max, \\infty}$ is an isomorphism.\n}\n\\hfill{$\\Box$}\n\\par\nWe next prove Theorem 1.1 for the case $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ being a coarse disjoint union of a sequence of finite metric spaces, by using Theorem 4.4.\nTo do so, we need the following lemma (cf. \\cite{GWY, OY, WiY2}).\n\\par\n{\\bf Lemma 4.5.} {\\it\nLet $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$ be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry. For each $d\\geq 0$, there is a short exact sequence\n$$0\\longrightarrow \\cK\\longrightarrow \\cmpdx \\longrightarrow \\cumpdxn \\longrightarrow 0$$\nsuch that the inclusion $\\cK\\longrightarrow \\cmpdx$ induces an injection on $K$-theory.\n}\n\\par\n{\\bf Proof.} Let $Z_d\\subset P_d(X)$ be a countable dense subset, and let $\\zdn=Z_d\\cap \\pdxn$ for all $d\\geq 0, \\nn$ as being used in Definition 3.1 and Definition 4.1.\nNote that $\\cK\\cong \\cK(\\ell^2(Z_d)\\otimes H_0)$ is an ideal of $\\cmpdx$.  There is a $*$-homomorphism\n$$\\Phi:\\; \\cpdx \\longrightarrow \\cupdxn$$\ndefined by $\\Phi(T)=[(\\Phi^{(0)}(T), \\cdots, \\Phi^{(n)}(T), \\cdots)]$ for $T\\in \\cpdx,$\nwith\n\\[\n\\Phi^{(n)}(T) = \\left\\{\n\\begin{array}{ll}\n0,                      & \\mbox{ if } n<N_R; \\\\\nT|_{\\zdn\\times \\zdn},   & \\mbox{ if } n\\geq N_R,\n\\end{array}\n\\right.\n\\]\nwhere $R=propagation(T)$ and $N_R\\in \\mathbb{N}$ is large enough such that\n$$d\\Big( X_n, \\, \\bigsqcup_{i=0}^{n-1}X_i \\Big)>2R$$\nfor all $n\\geq N_R$. The $*$-homomorphism $\\Phi$ extends to $C^*_{\\max}$-level\n$$\\Phi:\\; \\cmpdx \\longrightarrow \\cumpdxn$$\nsuch that $\\cK$ lives in the kernel of $\\Phi$. The induced $*$-homomorphism on the quotient\n$$\\Phi:\\; \\cmpdx\/\\cK \\longrightarrow \\cumpdxn$$\nhas an inverse $\\Psi$ defined as follows: for any\n$$\\ttt\\in \\cupdxn,$$\nlet $S=\\bigoplus_\\nn T^{(n)}$. Then $S\\in \\cpdx$ and $\\Phi(S+\\cK)=T$. Define\n$$\\Psi(T)=S+\\cK\\in \\cmpdx\/\\cK.$$\nThen $\\Psi$ extends to a $*$-homomorphism on $C^*_{\\max}$-level:\n$$\\Psi:\\; \\cumpdxn \\longrightarrow \\cmpdx\/\\cK$$\nwhich is the inverse of $\\Phi$. This gives the short exact sequence. The $K$-theory statement follows from \\cite{GWY, OY}.\n\\hfill{$\\Box$}\n\n\\par\n{\\bf Proposition 4.6.} {\\it\nLet $X=\\bigsqcup_{n\\in \\mathbb{N}} X_n$  be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry. If\n$X$ admits a fibred coarse embedding into Hilbert space, then the maximal coarse Baum-Connes conjecture holds for $X$. That is,\n$$ \\mu_{\\max}: \\lim_{d\\to \\infty} K_*(\\pdx) \\to  \\lim_{d\\to \\infty} K_*(\\cmpdx) \\cong K_*(\\cmx) $$\nis an isomorphism.\n}\n\\par\n{\\bf Proof.} For any $d\\geq 0$, there exists $N_d\\in \\mathbb{N}$ large enough such that $d(X_n, X_m)>d$ provided $n, m\\geq N_d$.  Let $X_{N_d}=\\bigcup_{n=0}^{N_d-1} X_n$.\nThen we have\n$$K_*(\\pdx)=K_*(P_d(X_{N_d}))\\bigoplus \\prod_{n=N_d}^\\infty K_*(\\pdxn).$$\nBy the definition of assembly maps and Lemma 4.5, we have the following commutative diagram:\n\n\\[\n\\xymatrix{ 0 \\ar[d]& 0 \\ar[d] \\\\\nK_*(P_d(X_{N_d}))\\oplus\\bigoplus_{n=N_d}^\\infty K_*(P_d(X_n)) \\ar[d] \\ar[r]& K_*(\\mathcal{K}) \\ar[d] \\\\\nK_*(P_d(X)) \\ar[d] \\ar[r]^{\\mu_{\\max} \\quad \\quad} & K_*(C_{max}^*(\\pdx)) \\ar[d]  \\\\\n\\frac{\\prod_{n=0}^\\infty K_*(\\pdxn)}{\\oplus_{n=0}^\\infty K_*(\\pdxn)}\n\\ar[r]^{\\mu_{\\max, \\infty}\\quad\\quad} \\ar[d] & K_*\\Big( \\cumpdxn  \\Big) \\ar[d] \\\\ 0 & 0}\n\\]\n\\par\nPassing to inductive limit as $d\\to \\infty$, the top horizontal arrow is an isomorphism for the following reason. An element in the sum, as a finite sequence, is supported on summands below some\nfixed $m$ and, as $d\\to \\infty$, will eventually be absorbed into the first term on a single simplex. Thus, the assertion reduces to the fact that the assembly map is an isomorphism for a bounded metric\nspace. Now by Theorem 4.4 together with the five lemma, we complete the proof.\n\\hfill{$\\Box$}\n\\par\nFinally, we are able to prove Theorem 1.1 by using Proposition 4.6.\n\\par\n{\\bf Proof of Theorem 1.1.} Fix a point $x_0\\in X$. For $n=0, 1, 2, \\cdots$, let\n$$X_n= \\big\\{ x\\in X \\; \\big| \\; n^3-n\\leq d(x, x_0) \\leq (n+1)^3+(n+1) \\big\\}$$\nand denote\n$$X^{(0)}=\\bigcup_{n:\\; even} X_n; \\quad \\quad X^{(1)}=\\bigcup_{n: \\; odd} X_n.$$\nThen $X=X^{(0)}\\cup X^{(1)}$, and each of $X^{(0)}$, $X^{(1)}$ and  $X^{(0)}\\cap X^{(1)}$ is the coarse disjoint union of a sequence of finite subspaces of $X$, which admits a fibred coarse\nembedding into Hilbert space as restricted from $X$.\n\\par\nMoreover, the pair $(X^{(0)}, X^{(1)})$ is ``$\\omega$-excisive'' (cf. \\cite{HRY}) in the sense that, for any $R>0$ there exists $S>0$\nsuch that\n$$\\Pen(X^{(0)}; R)\\cap \\Pen(X^{(1)}; R) \\subseteq \\Pen(X^{(0)}\\cap X^{(1)}; S),$$\nwhere $\\Pen(Y; R)=\\{x\\in X|d(x, Y)\\leq R\\}$ is the $R$-neighborhood of a subspace. Indeed,\nfor any $R>0$, take an integer $n_R>R$ and let $S=(n_R+1)^3$. Suppose $x\\in \\Pen(X^{(0)}; R)\\cap \\Pen(X^{(1)}; R)$. If $d(x, x_0)\\leq S$, then\n$x\\in \\Pen(X^{(0)}\\cap X^{(1)}; S)$ since $x_0\\in X^{(0)}\\cap X^{(1)}$. If $d(x, x_0)> S$, then there exist\n$$\\bar{x}\\in X^{(0)}-B_X(x_0, n_R^3), \\quad\\quad \\bar{y}\\in X^{(1)}-B_X(x_0, n_R^3) $$\nsuch that $d(x, \\bar{x})\\leq R$ and $d(x, \\bar{y})\\leq R$. Hence, $d(\\bar{x}, \\bar{y})\\leq 2R$. We claim that either $\\bar{x}\\in X^{(0)}\\cap X^{(1)}$, or $\\bar{y}\\in X^{(0)}\\cap X^{(1)}$. Otherwise, we would simultaneously have that\n$$\\bar{x}\\in \\bigcup_{n:even,\\, n\\geq n_R} \\big\\{ x\\in X\\, \\big|\\, n^3+n<d(x, x_0)<(n+1)^3-(n+1) \\big\\}$$\nand\n$$\\bar{y}\\in \\bigcup_{n:odd,\\, n\\geq n_R} \\big\\{ x\\in X\\, \\big|\\, n^3+n<d(x, x_0)<(n+1)^3-(n+1) \\big\\}.$$\nIt follows that $d(\\bar{x}, \\bar{y})\\geq 2\\, n_R>2R$, a contradiction. Consequently, either $\\bar{x}$ or $\\bar{y}$ is in $X^{(0)}\\cap X^{(1)}$, so that\n$x\\in \\Pen(X^{(0)}\\cap X^{(1)}; R) \\subset \\Pen(X^{(0)}\\cap X^{(1)}; S)$. Therefore, the pair $(X^{(0)}, X^{(1)})$ is ``$\\omega$-excisive'' .\n\\par\nFor each $d\\geq 0$, there exists $l_d>0$ such that, for any $x\\in X\\backslash B_X(x_0, l_d)$, the ball $B_X(x, d)$ is contained in either $X^{(0)}$ or $X^{(1)}$. Denote $K_d:=B_X(x_0, l_d)$. Then for Rips complexes, we have\n$$P_d(X)=P_d(K_d\\cup X^{(0)}) \\bigcup P_d(K_d\\cup X^{(1)}).$$\nThis is again an ``$\\omega$-excisive'' decomposition,  into subspaces which are coarsely equivalent to $X^{(0)}$ and $X^{(1)}$ respectively.  As a result (cf. \\cite{HRY}), we have the following\ncommutative diagram, in which the vertical arrows are assembly maps $\\mu_{\\max, \\infty}$ connecting two Mayer-Vietoris exact sequences-----the top on $K$-homology, whereas the bottom on $K$-theory:\n\n\\[\n\\xymatrix\n{\n  & AH_0 \\ar[rr] \\ar'[d][dd]     &  &  BH_0\\ar[rr]\\ar'[d][dd]  &  &  K_0\\Big(  \\pdx \\Big)  \\ar[dd]^{\\mu_{\\max}} \\ar[dl]        \\\\\n    K_1\\Big(  \\pdx \\Big) \\ar[ur]\\ar[dd]_{\\mu_{\\max}}    &  &  BH_1 \\ar[ll]\\ar[dd]     &  &  AH_1 \\ar[ll]\\ar[dd]          &    \\\\\n  & A_0 \\ar'[r][rr]              &  &  B_0 \\ar'[r][rr]         &  &  K_0\\Big(  \\cmx  \\Big) \\ar[dl]                       \\\\\n    K_1\\Big(  \\cmx  \\Big) \\ar[ur]                  &  &  B_1\\ar[ll]              &  &  A_1 \\ar[ll]                  &        }\n\\]\nWhere,\n$$AH_0= K_0\\Big(   P_d(K_d\\cup X^{(0)}) \\bigcap P_d(K_d\\cup X^{(1)}) \\Big) , \\quad BH_0= K_0\\Big(   P_d(K_d\\cup X^{(0)})  \\Big) \\bigoplus   K_0\\Big(   P_d(K_d\\cup X^{(1)})  \\Big) , $$\n$$A_0=K_0\\Big(  C^*_{\\max} ( X^{(0)}\\cap X^{1)})   \\Big),  \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad  B_0=K_0\\Big(  C^*_{\\max} ( X^{(0)}) \\Big) \\bigoplus K_0\\Big(  C^*_{\\max} ( X^{(1)})\\Big) ,$$\nand similarly for $K_1$. Passing to inductive limit as $d\\to \\infty$, clearly, Theorem 1.1 follows from (the proof of) Proposition 4.6 and the five lemma.\n\\hfill{$\\Box$}\n\n\n\n\n\n\\section{The twisted algebras at infinity}\nIn the rest of this paper, we shall prove that the evaluation homomorphism\n\\begin{small}\n$$e_*: \\lim_{d\\to \\infty} K_*\\Big( \\culmpdxn \\Big) \\longrightarrow \\lim_{d\\to \\infty} K_*\\Big( \\cumpdxn \\Big) $$\n\\end{small}\nis an isomorphism for a sequence of finite metric spaces $(X_n)_\\nn$ with uniform bounded geometry such that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding\ninto Hilbert space. It follows from the discussion of the last section that this suffices to complete the proof of Theorem 1.1.\n\\par\nIn this section, we shall introduce the twisted Roe algebras at infinity  \\\\\n$\\cumpdxna$ and their localization counterpart \\\\\n$\\culmpdxna$ for $(X_n)_\\nn$. Note that here the\ncoefficient algebras $\\cA(V_n)$ are defined on {\\em finite dimensional} affine subspaces $V_n$ of $H$. In section 6, we study various decompositions\nfor these twisted algebras to show that the evaluation map\n\\begin{small}\n$$e_*: \\lim_{d\\to \\infty} K_*\\Big( \\culmpdxna \\Big) \\longrightarrow \\lim_{d\\to \\infty} K_*\\Big( \\cumpdxna \\Big) $$\n\\end{small}\nfor the twisted algebras is an isomorphism. In section 7, we shall define the Bott maps $\\beta$, $\\beta_L$ and the Dirac maps $\\alpha$, $\\alpha_L$ to build the following\ncommutative diagram\n\\[\n\\xymatrix{\nK_*\\Big( \\culmpdxn \\Big) \\ar[r]^{\\quad e_*} \\ar[d]^{(\\beta_L)_*} & K_*\\Big( \\cumpdxn \\Big) \\ar[d]^{\\beta_*} \\\\\nK_*\\Big( \\culmpdxna \\Big)  \\ar@<1ex>[u]^{(\\alpha_L)_*} \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxna \\Big )\\;. \\ar@<1ex>[u]^{\\alpha_*}\n}\n\\]\nWe prove a geometric analogue of the Bott periodicity in finite dimensions, i.e. $\\alpha_*\\circ \\beta_*=identity$ and $(\\alpha_L)_*\\circ (\\beta_L)_*=identity$. Passing to the inductive limit\nas $d\\to \\infty$, a diagram chasing argument implies that the top evaluation map is an isomorphism,  as desired.\n\\par\n\\subsection*{{\\it \\S 5.1.  Preliminary}}\n Let $H$ be a separable Hilbert space. Denote by $V_a$, $V_b$ etc. the finite dimensional affine subspaces of $H$. Let $V_a^0$ be the linear subspace of $H$ consisting of differences of elements\nof $V_a$. Let $\\Cliff(V_a^0)$ be the complexified Clifford algebra of $V_a^0$ and $\\cC(V_a)$ the graded $C^*$-algebra of continuous functions vanishing at infinity from $V_a$ into $\\Cliff(V_a^0)$.\nLet $\\cS=C_0(\\mathbb{R})$, graded according to even and odd functions. Define the graded tensor product:\n$$\\cA(V_a)=\\cS\\widehat{\\otimes} \\cC(V_a).$$\nIf $V_a\\subseteq V_b$, then we have a decomposition $V_b=V_{ba}^0\\oplus V_a$, where $V_{ba}^0$ is the orthogonal complement of $V_a^0$ in $V_b^0$. For each $v_b\\in V_b$, we have a corresponding\ndecomposition $v_b=v_{ba}+v_a$, where $v_{ba}\\in V_{ba}^0$ and $v_a\\in V_a$. Every function $h$ on $V_a$ can be extended to a function $\\widetilde{h}$ on $V_b$ by the formula $\\widetilde{h}(v_{ba}+v_a)=h(v_a)$.\n\\par\n{\\bf Definition 5.1.} For affine subspaces $V_a\\subseteq V_b$, denote by $C_{ba}: V_b\\to \\Cliff(V_{ba}^0)$ the function $v_b\\mapsto v_{ba}\\in \\Cliff(V_{ba}^0)$, where $v_{ba}$ is considered as an element\nof $\\Cliff(V_{ba}^0)$ via the inclusion $V_{ba}^0\\subseteq \\Cliff(V_b^0)$. Let $X$ be the unbounded multiplier of $\\cS$ given by the function $t\\mapsto t$. Define a $*$-homomorphism\n$\\beta_{V_b, V_a}:\\cA(V_a)\\to \\cA(V_b)$, or simply denoted by $\\beta_{ba}$, by the formula\n$$\\beta_{ba}(g\\widehat\\otimes h)=g(X\\widehat\\otimes 1+ 1\\widehat\\otimes C_{ba})\\, (1\\widehat\\otimes \\wt h) $$\nfor all $g\\in \\cS, h\\in \\cC(V_a)$, where $g(X\\widehat\\otimes 1+1\\widehat\\otimes C_{ba})$ is defined by functional calculus.\n\\par\n{\\bf Definition 5.2.} If  $V_a\\subseteq V_b$, for any subset $O\\subset \\mathbb{R}_+\\times V_a$, define\n$$\\overline{O}^{\\beta_{ba}}=\\Big\\{ \\; \\big(t, \\;\\; v_{ba}+v_a \\big) \\in \\mathbb{R}_+\\times V_b \\; \\Big| \\;\\; \\Big( \\sqrt{t^2+\\|v_{ba}\\|^2}\\; ,\\;  v_a \\Big)\\in O \\; \\Big\\}.$$\n\\par\nFor any finite dimensional affine subspace $V_a$ of $H$, the algebra $C_0(\\mathbb{R}_+\\times V_a)$ is included in $\\cA(V_a)$ as its center. For any function $a\\in \\cA(V_a)$, the support of $a$,\ndenoted by $\\supp(a)$, is the complement of all points $(t, v)\\in \\mathbb{R}_+\\times V_a$ such that there exists $g\\in C_0(\\mathbb{R}_+\\times V_a)$ such that $g(t, v)\\not=0$ but $g\\cdot a=0$. Note\nthat if $V_a\\subseteq V_b$ and $a\\in \\cA(V_a)$, then\n$$\\supp\\big( \\beta_{ba}(a) \\big) = \\overline{\\supp(a)}^{\\beta_{ba}}.$$\n\\par\n{\\bf Definition 5.3.} Let $W_a, W_b, V_a, V_b, V_c$ be finite dimensional affine subspaces of $H$ with  $W_a\\subseteq W_b\\subset V_c$ and $V_a\\subseteq V_b\\subset V_c$.\nLet $t$ be an affine isometry from $W_b$ onto $V_b$ mapping $W_a$ onto $V_a$. Then $t$ canonically\ninduces a $*$-isomorphism from $\\cA(W_a)$ onto $\\cA(V_a)$. Note that $\\cA(V_a)$ is included in $\\cA(V_c)$ via $\\beta_{ca}$. We denote by $t_*$ the composition\n$$t_*: \\;\\; \\cA(W_a)\\stackrel{\\cong}{\\longrightarrow} \\cA(V_a) \\stackrel{\\beta_{ca}}{\\longrightarrow} \\cA(V_c).$$\nThen we have the following commutative diagram which is useful in the definition of product structure for the twisted Roe algebras.\n\\[\n\\xymatrix{\nt_*: & \\cA(W_a) \\ar[r]^{\\cong} \\ar[d]^{\\beta_{ba}} & \\cA(V_a) \\ar[r]^{\\beta_{ca}} \\ar[d]^{\\beta_{ba}}  & \\cA(V_c) \\ar[d]^{=}   \\\\\n& \\cA(W_b) \\ar[r]^{\\cong}                     & \\cA(V_b) \\ar[r]^{\\beta_{ca}} & \\cA(V_c)\n}\n\\]\n\\hfill{$\\Box$}\n\\par\n\\subsection*{\\it \\S 5.2. The twisted Roe algebras at infinity}\nLet $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry such that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding into Hilbert space.\nTo fix notations, we note that the notion of fibred coarse embedding into Hilbert space for the coarse disjoint union $X$ is equivalent to the following notion for the sequence\n$(X_n)_\\nn$:\n\\par\n{\\bf Definition 5.4.} A sequence of finite metric spaces $(X_n)\\nn$ with uniform bounded geometry is said to admit a fibred coarse embedding into Hilbert space if there exist\n\\begin{itemize}\n\\item a field of Hilbert spaces $(H_x)_{x\\in X_n, \\nn}$;\n\\item a section $s: X_n\\to\\displaystyle\\bigsqcup_{x\\in X_n} H_x$ for all $\\nn$;\n\\item two non-decreasing functions $\\rho_1$ and $\\rho_2$ from $[0, \\infty)$ to $(-\\infty, \\infty)$ with $\\lim_{r\\to \\infty}\\rho_i(r)=\\infty$ ($i=1, 2$);\n\\item a non-decreasing sequence of numbers $0\\leq \\l_0\\leq \\l_1\\leq \\cdots \\leq \\l_n\\leq \\cdots$ with $\\lim_{n\\to \\infty} \\l_n=\\infty$\n\\end{itemize}\nsuch that for each $x\\in X_n, \\nn$ there exists a trivialization\n$$t_x: (H_z)_{z\\in B(x, l_n)}\\longrightarrow B(x, l_n)\\times H$$\nsuch that the restriction  of $t_x$ to the fiber $H_z$ ($z\\in B(x, l_n)$) is an affine isometry $t_x(z): H_z\\to H$, satisfying\n\\begin{itemize}\n\\item[(1)] $\\rho_1(d(z, z'))\\leq \\|t_x(z)(s(z))-t_x(z')(s(z'))\\|\\leq \\rho_2(d(z, z'))$ for any $z, z'\\in B(x, l_n)$, $x\\in X_n, \\nn$;\n\\item[(2)] for any $x, y\\in X_n$ with $B(x, l_n)\\cap B(y, l_n)\\not=\\emptyset$, there exists an affine isometry $t_{xy}: H\\to H$ such that\n\t\t$t_{x}(z)\\circ t_{y}^{-1}(z)=t_{xy}$ for all $z\\in B(x, l_n)\\cap B(y, l_n)$.\n\\end{itemize}\n\\hfill{$\\Box$}\n\\par\nFor each $d\\geq 0$ and $\\nn$, let $\\pdxn$ be the Rips complex of $X_n$ at scale $d$ endowed with the spherical metric. For each $x\\in X_n$, denote by $\\Star(x)$ the open star of $x$ in\nthe second barycentric subdivision of $\\pdxn$. Take a countable dense subset $\\zdn\\subset \\pdxn$ for each $d\\geq 0$ in such a way that\n$$ (1) \\quad \\zdn\\subset \\displaystyle\\bigsqcup_{x\\in X_n} \\Star(x);  \\quad\\quad\\quad (2) \\quad \\zdn\\subseteq Z_{d', n}  \\mbox{ when }  d<d'. $$\n\\par\nFor any $x\\in \\zdn$, there exists a unique point $\\bar{x}\\in X_n$ such that $x\\in \\Star(\\bar{x})$. We define\n$$H_x=H_{\\bar{x}}, \\quad\\quad s(x)=s(\\bar{x})$$\nfor all $x\\in \\zdn\\cap \\Star(\\bar x)$ and let\n$$t_x(z)=t_{\\bar{x}}(\\bar{z}):\\; H_z\\to H$$\nfor all $x, z\\in \\zdn, \\; \\nn$ with $\\bar z\\in B(\\bar x, l_n)$.\n\\par\nFor each $\\nn$, define $V_n$ to be the {\\em finite dimensional} affine subspace of $H$ spanned by $t_x(z)(s(z))$ for all $z\\in B(x, l_n), \\; x\\in X_n$:\n$$V_n :=\\mbox{ affine-span } \\Big\\{ \\; t_x(z)(s(z))\\; \\Big| \\; z\\in B(x, l_n), \\; x\\in X_n \\Big\\} \\,.$$\nFor each $x\\in \\zdn$, $k\\geq 0$, define\n$$W_k(x)=\\mbox{ affine-span } \\Big\\{ \\; t_x(z)(s(z))\\; \\Big| \\; z\\in B_{\\pdxn} (x, k)  \\; \\Big\\} \\subseteq V_n \\,.$$\nNote that for each $k\\geq 0$, there exists $N\\in \\mathbb{N}$ such that  $W_k(x)$ is well-defined for $x\\in \\zdn$ with $n\\geq N$, and is an affine subspace of $V_n$. By Definition 5.1,\nthe inclusion $W_k(x)\\subseteq V_n$ induces the inclusion\n$$\\beta_{V_n, W_k(x)}:\\; \\cA(W_k(x))\\to \\cA(V_n).$$\n\\par\n{\\bf Definition 5.5.} For each $d\\geq 0$, define $\\cupdxna$ to be the set of all equivalence classes $\\ttt$ of sequences $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ described as follows:\n\\begin{itemize}\n\\item[(1)] $T^{(n)}$ is a bounded function from $\\zdn \\times \\zdn$ to $\\cA(V_n)\\widehat\\otimes \\cK$ for all $\\nn$;\n\\item[(2)] for any bounded subset $B\\subset \\pdxn$, the set\n\t\t\t$$\\{ (x, y)\\in B\\times B \\cap \\zdn\\times \\zdn \\; | \\; T^{(n)} (x, y)\\not= 0 \\}$$\n\t\tis finite;\n\\item[(3)] there exists $L>0$ such that\n\t\t\t$$\\#\\{y\\in \\zdn \\; | \\; T^{(n)}(x, y)\\not= 0 \\}<L, \\quad \\#\\{y\\in \\zdn \\; | \\; T^{(n)}(y, x)\\not= 0 \\}<L $$\n\t\t\tfor all $x\\in \\zdn,\\; \\nn$;\n\\item[(4)] there exists $R>0$ such that $T^{(n)}(x, y)=0$ whenever $d(x, y)>R$  for $x, y\\in \\zdn,\\; \\nn$; (the least such $R$ is call the propagation of the sequence\n\t\t\t $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ .)\n\\item[(5)] there exists $r>0$ such that\n\t\t\t\t\\[\n\t\t\t\t\t\\begin{array}{rcl}\n\t\t\t\t\t\t\\supp(T^{(n)}(x, y) )  & \\subseteq & B_{\\mathbb{R}_+\\times V_n} \\big( t_x(x)(s(x)), \\; r\\big)     \\\\\n\t\t\t\t\t\t\t\t\t\t  & :=   & \\big\\{ (\\tau, v)\\in \\mathbb{R}_+\\times V_n \\; \\big| \\; \\tau^2+\\|v-t_x(x)(s(x))\\|^2<r^2 \\big\\}\n\t\t\t\t\t\\end{array}\n\t\t\t\t\\]\n\t\tfor all $x, y\\in \\zdn, \\; \\nn$.\n\\item[(6)] there exist $k\\geq 0$ and $K>0$ depending only on the sequence $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ (not on $n$) such that, for each $x, y\\in \\zdn$, there exists\n\t\t\t$$T_1^{(n)}(x, y)\\in \\cA(W_k(x))\\widehat\\otimes \\cK\\cong \\cS\\widehat\\otimes \\cC(W_k(x))\\widehat\\otimes \\cK$$\n\t\t\tof the form $\\sum_{i=1}^K g_i\\widehat\\otimes h_i\\widehat\\otimes k_i$ where $g_i\\in \\cS$, $h_i\\in \\cC(W_k(x))$,\n\t\t\t$k_i\\in \\cK$ for $1 \\leq i \\leq K$  such that\n\t\t\t\t$$ T^{(n)}(x, y) = \\Big( \\beta_{V_n, W_k(x)} \\widehat\\otimes 1 \\Big) \\big( T^{(n)}_1 (x, y) \\big) \\,.$$\n\\item[(7)] there exists $c>0$ such that if $T_1^{(n)}(x, y)\\in \\cA(W_k(x))\\widehat\\otimes \\cK$  as above, and $w\\in \\mathbb{R}_+\\times W_k(x)$ is of norm one, then\n\t\tthe derivative of $T^{(n)}_1(x, y)$ in the direction $w$, $\\nabla_w\\big( T^{(n)}_1(x, y) \\big)$, exists in $\\cA(W_k(x))\\widehat\\otimes \\cK$ and is of norm at most $c$.\n\\end{itemize}\nThe equivalence relation $\\sim$ on these sequences is defined by\n$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots) \\sim (S^{(0)}, \\cdots, S^{(n)}, \\cdots)$$\nif and only if\n$$\\lim_{n\\to \\infty} \\sup_{x, y\\in \\zdn} \\big\\| T^{(n)}(x, y) - S^{(n)}(x, y) \\big\\|_{\\cA(V_n)\\widehat\\otimes \\cK} =0.$$\nThe product structure for $\\cupdxna$ is defined as follows. For any two elements $\\ttt$ and $S=\\big[ (S^{(0)}, \\cdots, S^{(n)}, \\cdots) \\big]$ in $\\cupdxna$, their product is defined to be\n$$\\big[ \\big( (TS)^{(0)}, \\cdots, (TS)^{(n)}, \\cdots \\big) \\big] \\,,$$\nwhere there exists a sufficiently large $N\\in \\mathbb{N}$ depending on the propagation of the two representative sequences, such that\n$ (TS)^{(n)}=0$ for all $n=0, 1, 2, \\cdots, N-1$ and\n$$(TS)^{(n)}(x, y)=\\sum_{z\\in \\zdn} \\Big( T^{(n)}(x, z) \\Big) \\cdot \\Big( \\big(t_{xz}\\big)_* \\big( S^{(n)}(z, y) \\big) \\Big)$$\nfor all $x, y\\in \\zdn, \\; n\\geq N$.\n\\par\n{\\bf Remark 5.6.} Some explanations to the above formula are given here. Note that there exists $k\\geq 0$ such that\n\t\t\t\t$ S^{(n)}(z, y) = \\Big( \\beta_{V_n, W_k(z)} \\widehat\\otimes 1 \\Big) \\big( S^{(n)}_1 (z, y) \\big)$ for $z, y\\in \\zdn, \\;\\nn$, where\n\t\t\t\t$S_1^{(n)}(z, y)\\in \\cA(W_k(z))\\widehat\\otimes \\cK$.  For the propagation $R$ of the representative sequence $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big)$ and the above $k$, there\nexists $N\\in \\mathbb{N}$ large enough such that for any $n\\geq N$, there exists $\\widetilde{l_n}>0$ (since $\\pdxn$ is coarsely equivalent to $X_n$)\nsuch that the trivialization\n$$t_x: \\;\\; \\Big\\{ H_z \\Big\\}_{z\\in B_{\\pdxn}(x, \\widetilde{l_n})} \\longrightarrow  B_{\\pdxn}(x, \\widetilde{l_n})\\times H$$\nmakes sense on $B_{\\pdxn}(x, \\widetilde{l_n})$, and $R+k<\\widetilde{l_n}$. If $d(x, z)\\leq R$ in $\\pdxn$, then the affine isometry $t_{xz}=t_x(w)\\circ t_z^{-1}(w):\\; H\\to H$ for all\n$$w\\in B_{\\pdxn}(x, \\widetilde{l_n})\\cap B_{\\pdxn}(z, \\widetilde{l_n}).$$\nNote that $t_{xz}$ maps the affine subspace\n$$W_k(z)=\\mbox{ affine-span } \\Big\\{ t_z(w)(s(w)) \\; \\Big| \\; w\\in B_{\\pdxn}(z, k) \\cap \\zdn \\Big\\}$$\nonto an affine subspace of $W_{R+k}(x)$ since $B_{\\pdxn}(z, k)\\subseteq B_{\\pdxn}(x, R+k)$. By Definition 5.3, the composition\n$$t_{xz}: W_k(z)\\to t_{xz}\\big( W_k(z) \\big) \\subseteq W_{R+k}(x) \\subseteq V_n$$\ninduces the $*$-homomorphism\n$$\\big(t_{xz}\\big)_*:\\; \\cA( W_k(z) ) \\to \\cA(V_n).$$\nWe define\n$$\\big(t_{xz}\\big)_*\\Big( S^{(n)}(z, y) \\Big):= \\big(t_{xz}\\big)_*\\Big( S_1^{(n)}(z, y) \\Big)$$\nin the above product formula. Note that for $\\nn$ large enough, this definition of $\\big(t_{xz}\\big)_*\\Big( S^{(n)}(z, y) \\Big)$ does not depend on the choice of $k$\n(see Definition 5.3, and \\cite{HG, HKT}).\n\\hfill{$\\Box$}\n\\par\nThe $*$-structure for $\\cupdxna$ is defined by the formula\n$$ \\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]^*=   \\big[ \\big( (T^*)^{(0)}, \\cdots, (T^*)^{(n)}, \\cdots \\big) \\big] \\,,$$\nwhere\n$$(T^*)^{(n)}(x, y)=\\big( t_{xy} \\big)_* \\Big( \\big( T^{(n)}(y, x)\\big)^*\\Big)$$\nfor all but finitely many $n$, and $0$ otherwise.\n\\par\nNow, $\\cupdxna$ is made into a $*$-algebra by using the additional usual matrix operations. Define $\\cumpdxna$ to be the completion of $\\cupdxna$ with respect to the norm\n\\begin{footnotesize}\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cupdxna \\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\end{footnotesize}\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 5.7.}  For each $d\\geq 0$, define $\\culpdxna$  to be the $*$-algebra of all bounded and uniformly norm-continuous functions\n$$f: \\; [0,\\infty)\\longrightarrow \\cupdxna$$\nsuch that $f(t)$ is of the form $f(t)=[(f^{(0)}(t), \\cdots, f^{(n)}(t), \\cdots )]$ for all $t\\in [0, \\infty)$, where the family of functions\n$(f^{(n)}(t))_{\\nn, t\\geq 0}$ satisfy the conditions in Definition 5.5 with uniform constants, and there exists a bounded function $R:\\; [0, \\infty)\\to [0,\\infty)$ with\n$\\lim_{t\\to \\infty} R(t)=0$ such that\n$$ \\big( f^{(n)}(t)\\big) (x, y) =0  \\quad \\mbox{whenever} \\quad d(x, y)>R(t)$$\nfor all $x, y\\in \\zdn$, $\\nn$, $t\\in [0, \\infty)$.\n\\par\nDefine $\\culmpdxna$ to be the completion of\n$$\\culpdxna$$\nwith respect to the norm\n$$\\big\\| f \\big\\|_{\\max} :=\\sup_{t\\in [0, \\infty)} \\big\\|f(t)\\big\\|_{\\max}.$$\n\\hfill{$\\Box$}\n\\par\nThe evaluation homomorphism\n$$e: \\; \\culmpdxna \\longrightarrow \\cumpdxna $$\ndefined by $e(f)=f(0)$ induces the evaluation homomorphism on $K$-theory:\n\\begin{small}\n$$ \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxna  \\big) \\stackrel{e_*}{\\longrightarrow}   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxna  \\big).$$\n\\end{small}\n\n\n\n\\section{Decompositions of twisted algebras}\nIn this section we shall prove the following result.\n\\par\n{\\bf Theorem 6.1.} {\\it\nLet $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry which admit a fibred coarse embedding into Hilbert space. Then\nthe evaluation homomorphism\n\\begin{small}\n$$ \\lim_{d\\to\\infty}       K_*\\big(   \\culmpdxna  \\big) \\stackrel{e_*}{\\longrightarrow}   \\lim_{d\\to\\infty}     K_*\\big(   \\cumpdxna  \\big).$$\n\\end{small}\nis an isomorphism.\n}\n\\par\nThe proof proceeds by decomposing the twisted algebras into various smaller subalgebras and applying a Mayer-Vietoris argument.\n\\subsection*{\\it \\S 6.1. Coherent system of open subsets of $\\mathbb{R}_+\\times V_n$ ($\\nn$)}\nTo begin with, we shall discuss ideals of the twisted algebras supported on certain open subsets.\n\\par\n{\\bf Definition 6.2.} A collection $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ of open subsets of $\\mathbb{R}_+\\times V_n, \\; \\nn,$ is said to be a {\\em coherent system} if for all but finitely many $\\nn$,\nthe following conditions hold:\n\\begin{itemize}\n\\item[(1)] for any subset $C\\subseteq B(x, l_n)\\cap B(y, l_n)$ with $x, y\\in X_n$, we have\n\t\t$$ O_{n, x}\\cap \\big(\\mathbb{R}_+\\times W_C(x) \\big) = t_{xy} \\Big( O_{n, y} \\cap \\big(\\mathbb{R}_+\\times W_C(y) \\big) \\Big),$$\n\t\twhere\n\t\t$$W_C(x)=\\mbox{ affine-span } \\Big\\{ t_x(z)(s(z)) \\; \\Big| \\; z\\in C \\Big\\}=t_{xy}\\big( W_C(y) \\big),$$\n\t\t$$W_C(y)=\\mbox{ affine-span } \\Big\\{ t_y(z)(s(z)) \\; \\Big| \\; z\\in C \\Big\\}=t_{yx}\\big( W_C(x) \\big),$$\n\t\tand $t_{xy}=t_x(z)\\circ t_y^{-1}(z):\\; H\\to H$ for all $z\\in B(x, l_n)\\cap B(y, l_n)$.\n\\item[(2)] for any $C\\subseteq B(x, l_n)$, where $x\\in X_n, \\;\\nn$, and any affine subspace $W$ with $W_C(x)\\subseteq W\\subseteq V_n$, we have\n\t\t$$ \\overline{O_{n, x}\\cap \\big(\\mathbb{R}_+\\times W_C(x) \\big)}^{\\beta_{W, W_C(x)}} \\subseteq  O_{n, x}\\cap \\big(\\mathbb{R}_+\\times W \\big),$$\n\t\twhere recall (Definition 5.3) that for $V_a\\subseteq V_b$ and $O\\subseteq \\mathbb{R}_+\\times V_a$,\n\t\t$$ \\overline{O}^{\\beta_{ba}} = \\big\\{  (t, v_{ba}+v_a)\\in \\mathbb{R}_+\\times V_b \\; \\big| \\; \\big( \\sqrt{t^2+\\|v_{ba}\\|^2} \\; , \\;v_a \\big)\\in O \\big\\}.$$\n\\end{itemize}\n \\hfill{$\\Box$}\n\\par\n{\\bf Definition 6.3.} Let $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ be a coherent system of open subsets of $\\mathbb{R}_+\\times V_n, \\; \\nn$. For each $d\\geq 0$, define\n$$\\cupdxna_O$$\nto be the $*$-subalgebra of $\\cupdxna$ generated by the equivalence classes of those sequences $\\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]$ such that\n$$\\supp (T^{(n)}(x, y)) \\subseteq O_{n, \\bar{x}}$$\nfor all $x, y\\in \\zdn$ with $x\\in \\Star(\\bar x)$ for all $n\\geq N$ for some $n\\in \\mathbb{N}$ large enough depending on the sequence  $\\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]$.\n\\par\nDefine $\\cumpdxna_O$ to be the norm closure of  \\\\\n$\\cupdxna_O$ in $\\cumpdxna$.\n \\hfill{$\\Box$}\n\\par\n{\\bf Lemma 6.4.} {\\it\n$\\cumpdxna_O$ is a two sided ideal of    \\\\\n$\\cumpdxna$.\n}\n\\par\n{\\bf Proof.} Suppose\n$$\\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]\\in \\cupdxna  \\,,$$\n$$\\big[ \\big( S^{(0)}, \\cdots, S^{(n)}, \\cdots \\big) \\big]\\in \\cupdxna_O  \\,,$$\nso that (1) there exists $R>0$ such that the propagation of $T^{(n)}$ and $S^{(n)}$ is at most $R$ for all $\\nn$; (2) $\\supp(S^{(n)}(z, y))\\subseteq O_{n,z}$ for all $z, y\\in \\zdn, \\,\\nn$;\n(3) there exists $k\\geq 0$ such that $S^{(n)}(z, y)=\\big( \\beta_{V_n, W_k(z)}\\widehat\\otimes 1\\big) \\big( S_1^{(n)}(z, y) \\big)$ for some $S_1^{(n)}(z, y)\\in \\cA(W_k(z))\\widehat\\otimes \\cK$\nfor all $z, y \\in \\zdn \\,\\nn$. It follows that\n$$\\supp \\big(S_1^{(n)}(z, y) \\big) \\subseteq O_{n, z}\\cap \\big( \\mathbb{R}_+\\times W_k(z) \\big), $$\nand\n\\[\n\\begin{array}{rcl}\n\\supp\\Big( \\big( t_{xz} \\big)_* \\big( S_1^{(n)}(z, y) \\big) \\Big) & \\subseteq & t_{xz} \\big(  O_{n, z}\\cap (\\mathbb{R}_+\\times W_k(z)) \\big)     \\\\\n  & = & O_{n, x}\\cap \\big( \\mathbb{R}_+\\times W_{B(z, k)}(x) \\big)  \\,,\n\\end{array}\n\\]\nwhere $B(z, k):=B_{\\pdxn}(z, k)$ and\n$$W_k(z)=\\mbox{ affine-span }\\Big\\{ t_z(w)(s(w)) \\; \\Big| \\; w\\in B_{\\pdxn}(z, k) \\cap \\zdn \\Big\\},$$\n$$W_{B(z, k)}(x) = t_{xz}(W_k(z)).$$\nIt follows from the definition of a coherent system that\n\\[\n\\begin{array}{rcl}\n\\supp\\Big( \\big( t_{xz} \\big)_* \\big( S^{(n)}(z, y) \\big) \\Big) & \\subseteq & \\overline{ \\Big(  O_{n, x}\\cap (\\mathbb{R}_+\\times W_{B(z, k)}(x)) \\Big) } ^{\\beta_{V_n, W_{B(z, k)}(x)}}    \\\\\n  & \\subseteq  & O_{n, x} \\,.\n\\end{array}\n\\]\nHence,\n$$\\supp\\Big( T^{(n)}(x, z) \\cdot \\big( t_{xz} \\big)_* \\big( S^{(n)}(z, y) \\big) \\Big) \\subseteq O_{n, x}, $$\nfor all but finitely many $\\nn$. That is,\n$$\\big[ \\big( (TS)^{(0)}, \\cdots, (TS)^{(n)}, \\cdots \\big) \\big]\\in \\cupdxna_O.$$\nThe proof is complete.\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 6.5.} Let $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ be a coherent system of open subsets of $\\mathbb{R}_+\\times V_n, \\; \\nn$. For each $d\\geq 0$, define\n$$\\culpdxna_O$$\nto be the $*$-subalgebra of $\\culpdxna$ consisting of all functions\n$$f:\\; [0, \\infty) \\longrightarrow \\cupdxna_O.$$\nDefine $\\culmpdxna_O$ to be the completion of    \\\\\n$\\culpdxna_O$ in $\\culmpdxna$.\n\\hfill{$\\Box$}\n\\par\nNote that $\\culmpdxna_O$ is an ideal of    \\\\\n$\\culmpdxna$. We also have an  evaluation homomorphism\n$$e: \\; \\culmpdxna_O\\rightarrow \\cumpdxna_O$$\ngiven by $e(f)=f(0)$.\n\\par\n\n\\subsection*{\\it \\S  6.2. Strong Lipschitz homotopy invariance}\nIn this subsection, we shall investigate ideals of the twisted algebras supported on those coherent systems which are separated by subsets of $X_n$, $\\nn$. These ideals can be decomposed into certain algebras\ndefined on uniformly bounded subsets of $P_d(X_n)$, $\\nn$, whose $K$-theory turns out to be strongly Lipschitz homotopy invariant. As a result, the evaluation homomorphism for these ideals is\nan isomorphism. This  constitutes a major step towards the proof of Theorem 6.1.\n\\par\nLet $\\Gamma_n$ be a subset of $X_n$ for each $\\nn$ and denote $\\Gamma=(\\Gamma_n)_\\nn$. Let $r>0$.\n\\par\n{\\bf Definition 6.6.} A coherent system $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ of open subsets of  $\\mathbb{R}_+\\times V_n, \\; \\nn,$ is said to be {\\em $(\\Gamma, r)$-separate} if\nthere exist open subsets $\\big( O_{n, x, \\gamma} \\big)_{\\gamma\\in \\Gamma_n\\cap B(x, l_n)}$ of $\\mathbb{R}_+\\times V_n$ for all $x\\in X_n,\\, \\nn$, such that\n\\begin{itemize}\n\\item $O_{n, x}=\\displaystyle\\bigcup_{\\gamma\\in \\Gamma_n\\cap B(x, l_n)} O_{n, x, \\gamma} $ ;\n\\item $ O_{n, x, \\gamma} \\cap O_{n, x, \\gamma'} =\\emptyset $ for distinct $\\gamma, \\gamma'\\in \\Gamma_n\\cap B(x, l_n)$ ;\n\\item $ O_{n, x, \\gamma} \\subseteq B_{\\mathbb{R}_+\\times V_n}\\Big( t_x(\\gamma)(s(\\gamma)) \\, ,  \\, r \\Big)$ for each $\\gamma\\in \\Gamma_n\\cap B(x, l_n)$.\n\\end{itemize}\n\\hfill{$\\Box$}\n\\par\n{\\bf Theorem 6.7.} {\\it\nIf a coherent system $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ is  $(\\Gamma, r)$-separate, then the evaluation homomorphism on $K$-theory\n\\[\n\\begin{array}{ll}\ne_*: &   \\displaystyle \\lim_{d\\to \\infty} K_*\\Big(  \\culmpdxna_{O} \\Big)             \\\\\n&   \\longrightarrow  \\displaystyle \\lim_{d\\to \\infty} K_*\\Big(  \\cumpdxna_{O} \\Big)\n\\end{array}\n\\]\nis an isomorphism.\n}\n\\par\nWe need some preparations before we can prove Theorem 6.8. Suppose $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ is  $(\\Gamma, r)$-separate for some  $\\Gamma=(\\Gamma_n)_\\nn$ and $r>0$ as above.\nFor $d\\geq 0$, let $(Y_\\gamma)_{\\gamma\\in \\Gamma_n}$ be a collection of closed subsets of $\\pdxn$ for each $\\nn$ such that\n\\begin{itemize}\n\\item $(Y_\\gamma)_{\\gamma\\in \\Gamma_n, \\,\\nn}$ is uniformly bounded, i.e. there exists $K>0$ such that $diameter(Y_\\gamma)\\leq K$ for all $\\gamma\\in \\Gamma_n, \\nn$.\n\\item $\\gamma\\in Y_\\gamma$ for all $\\gamma\\in \\Gamma_n, \\nn$.\n\\end{itemize}\nIn particular, we are mainly concerned with the following cases:\n\\begin{itemize}\n\\item[(1)] $Y_\\gamma=B(\\gamma, S):=\\big\\{ x\\in \\pdxn \\;\\big| \\; d(x, \\gamma)\\leq S \\big\\}$ for some common $S>0$ for all $\\gamma\\in \\Gamma_n, ,\\nn$.\n\\item[(2)] $Y_\\gamma=\\Delta_\\gamma(S)$, the simplex in $\\pdxn$ with vertices $\\{x\\in X_n | d(x, \\gamma)\\leq S \\}$ for some common $S>0$ for all $\\gamma\\in \\Gamma_n, \\,\\nn$.\n\\item[(3)] $Y_\\gamma=\\{\\gamma\\}$ for all $\\gamma\\in \\Gamma_n, \\,\\nn$.\n\\end{itemize}\n\\par\n{\\bf Definition 6.8.} For an open subset $O\\subseteq \\mathbb{R}_+\\times V_n$, we denote by $\\cA(O)$ the $C^*$-subalgebra of $\\cA(V_n)$ generated by the functions whose supports are\ncontained in $O$. Note that $\\cA(O)$ is an ideal of $\\cA(V_n)$.\n\\par\n{\\bf Definition 6.9.}  Define $A_\\infty[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$ to be the $*$-subalgebra of\n\\begin{equation}\\label{qqq}\n\\frac\n{\\prod_\\nn \\Big( \\bigoplus_{\\gamma\\in \\Gamma_n}  C^*_{\\max}(Y_\\gamma)\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\Big) }\n{\\bigoplus_\\nn \\Big( \\bigoplus_{\\gamma\\in \\Gamma_n}  C^*_{\\max}(Y_\\gamma)\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\Big) }\n\\end{equation}\nconsisting of elements of the form $\\big[ \\big( T^{(0)}, \\cdots, T^{(n)},  \\cdots \\big) \\big]$ where\n$$T^{(n)}=\\bigoplus_{\\gamma_\\in \\Gamma_n} T^{(n)}_\\gamma$$\nwith\n$$T^{(n)}_\\gamma\\in \\mathbb{C}[Y_\\gamma]\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\subset C^*_{\\max}(Y_\\gamma)\\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}),$$\nand when viewed as functions\n$$T^{(n)}_\\gamma:\\; (\\zdn\\times \\zdn)\\cap (Y_\\gamma\\times Y_\\gamma) \\rightarrow \\cA(O_{n, \\gamma, \\gamma}),$$\nthe family  $(T^{(n)}_\\gamma)_{\\gamma\\in \\Gamma_n, \\nn}$ satisfy the conditions in Definition 5.5 with uniform constants.\n\\par\nDefine $A^*_\\infty ( \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn )$ to be the completion of\n$$A_\\infty[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$$\ninside the $C^*$-algebra (1).\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 6.10.} Define $A_{L, \\infty}[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$ to be the $*$-algebra of all bounded and uniformly norm-continuous functions\n$$f: \\; [0,\\infty)\\longrightarrow A_\\infty[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$$\nwhere $f(t)$ is of the form $f(t)=[(f^{(0)}(t), \\cdots, f^{(n)}(t), \\cdots )]$ for all $t\\in [0, \\infty)$ and\n$$f^{(n)}(t)=\\bigoplus_{\\gamma_\\in \\Gamma_n} f^{(n)}_\\gamma (t)$$\nsuch that the family of functions\n$(f_\\gamma^{(n)}(t))_{\\gamma\\in \\Gamma_n, \\nn, t\\geq 0}$ satisfy the conditions in Definition 5.5 with uniform constants, and there exists a bounded function $R:\\; [0, \\infty)\\to [0,\\infty)$ with\n$\\lim_{t\\to \\infty} R(t)=0$ such that\n$$ \\big( f_\\gamma^{(n)}(t)\\big) (x, y) =0  \\quad \\mbox{whenever} \\quad d(x, y)>R(t)$$\nfor all $x, y\\in \\zdn\\cap Y_\\gamma$, $\\gamma\\in \\Gamma_n$, $\\nn$, $t\\in [0, \\infty)$.\n\\par\nDefine $A^*_{L, \\infty} \\big( \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn \\big)$ to be the completion of\n$$A_{L, \\infty}[ \\, ( Y_\\gamma: \\; \\gamma\\in \\Gamma_n)_\\nn]$$\nwith respect to the norm\n$$\\big\\| f \\big\\|_{\\max} :=\\sup_{t\\in [0, \\infty)} \\big\\|f(t)\\big\\|_{\\max}.$$\n\\hfill{$\\Box$}\n\\par\n{\\bf Proposition 6.11.} {\\it\nSuppose a coherent system $O=(O_{n, x})_{x\\in X_n, \\,\\nn}$ is  $(\\Gamma, r)$-separate for some  $\\Gamma=(\\Gamma_n)_\\nn$ and $r>0$ as above. Then\n\\begin{itemize}\n\\item  $\\displaystyle  \\cumpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{\\infty}\\big( \\,\\big( B(\\gamma, S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\, ,$\n\\item  $\\displaystyle  \\culmpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{L, \\infty}\\big( \\,\\big( B(\\gamma, S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\, ,$\n\\item  $\\displaystyle \\lim_{d\\to \\infty}\\cumpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{\\infty}\\big( \\,\\big( \\Delta_\\gamma (S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\, ,$\n\\item  $\\displaystyle  \\lim_{d\\to \\infty}\\culmpdxna_O \\cong \\lim_{S\\to \\infty} A^*_{L, \\infty}\\big( \\,\\big( \\Delta_\\gamma (S);  \\; \\gamma\\in \\Gamma_n  \\big)_\\nn \\big)  \\,. $\n\\end{itemize}\n}\n\\par\n{\\bf Proof.} We shall establish an isomorphism for the first item. Take arbitrarily an element\n$$\\ttt\\in \\cupdxna_O,$$\nwhere the functions $T^{(n)}:\\; \\zdn\\times \\zdn\\rightarrow \\cA(V_n)\\widehat\\otimes \\cK$ have supports\n$$\\supp\\big( T^{(n)}(x, y) \\big) \\subseteq O_{n, x} \\subseteq \\bigsqcup_{\\gamma\\in \\Gamma_n\\cap B_{X_n}(x, l_n)} O_{n, x, \\gamma}$$\nfor all $x, y\\in \\zdn, \\nn, $ since the coherent system $O$ is $(\\Gamma, r)$-separate. Then we have a direct sum decomposition\n$$T^{(n)}(x, y)=\\bigoplus_{\\gamma\\in \\Gamma_n\\cap B_{X_n}(x, l_n)}  T_\\gamma^{(n)}(x, y) \\,,$$\nwhere\n$$T_\\gamma^{(n)}(x, y) = T^{(n)}(x, y)\\Big|_{O_{n, x, \\gamma}}  \\in \\cA(O_{n, x, \\gamma})\\widehat\\otimes \\cK $$\nis the restriction on a subset for all  $x, y\\in \\zdn, \\nn, $ and $\\gamma\\in B_{X_n}(x, l_n)\\cap \\Gamma_n$.  On the other hand, it follows from the support condition (5) in Definition 5.5, there\nexists $\\bar r>0$ such that\n$$\\supp\\big( T^{(n)}(x, y) \\big) \\subseteq B_{\\mathbb{R}_+\\times V_n} \\big(  t_x(x)(s(x)) \\,, \\, \\bar{r} \\big) \\,.$$\nHence, $T_\\gamma^{(n)}(x, y) =0$ whenever\n$$d\\Big( t_x(\\gamma)(s(\\gamma)) \\,, \\, t_x(x)(s(x)) \\Big) > \\bar{r}+r$$\nin $V_n$. It follows that there exists $S>0$ (since the section $s$ is locally a coarse embedding) such that $T_\\gamma^{(n)}(x, y) =0$ whenever $d(x, \\gamma)>S$ for all\n$x, y\\in \\zdn, \\gamma\\in \\Gamma_n,  \\nn $. Now define\n\\begin{equation}\\label{rrr}\nS_\\gamma^{(n)}(x, y)=\\big( t_{\\gamma  x} \\big)_*\\big( T_\\gamma^{(n)}(x, y) \\big) \\,.\n\\end{equation}\nSince $\\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots  \\big)$ has finite propagation, there exists $N>0$ large enough such that, if $n\\geq N$, then $S_\\gamma^{(n)}(x, y)$ is well-defined, and\n$$S_\\gamma^{(n)}(x, y)\\in \\cA(O_{n, \\gamma, \\gamma})\\widehat\\otimes \\cK$$\n  for all $x, y\\in \\zdn\\cap B_{\\pdxn}(\\gamma, S)$ with $\\gamma\\in \\Gamma_n\\cap B_{X_n}(x, l_n)$ and $n\\geq N$.\nTherefore, if we write $B(\\gamma, S)=B_{\\pdxn}(\\gamma, S)$ and view a function as a matrix, we have\n\\[\n\\begin{array}{rcl}\nS_\\gamma^{(n)}  &  =  &  \\Big[ S_\\gamma^{(n)}(x, y)  \\Big] _ {(x, y)\\in (\\zdn\\times \\zdn) \\cap (B(\\gamma, S)\\times B(\\gamma, S))}  \\\\\n  & \\in & \\mathbb{C}\\big[ B(\\gamma, S)  \\big]   \\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma})  \\\\\n  & \\subseteq   & C^*_{\\max}\\big( B(\\gamma, S) \\big)  \\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma}) \\,.\n\\end{array}\n\\]\nDefine $S^{(n)}=0$ for all $n\\leq N-1$ and define\n$$S^{(n)}=\\bigoplus_{\\gamma\\in \\Gamma_n} S_\\gamma^{(n)} \\in  \\bigoplus_{\\gamma\\in \\Gamma_n} C^*_{\\max}\\big( B(\\gamma, S) \\big)  \\widehat\\otimes \\cA(O_{n, \\gamma, \\gamma})$$\nfor $n\\geq N$. Then the class $S=[( S^{(0)}, \\cdots, S^{(n)}, \\cdots )]$ is an element of\n$$A_{\\infty}\\big[ \\, \\big( B(\\gamma, S); \\, \\gamma\\in \\Gamma_n \\big)_\\nn \\big] \\, .$$\nNow the correspondence $T\\mapsto S$ extends to a $*$-isomorphism (see also the proof of Lemma 3.9 in \\cite{SpWi} for essentially the same arguments which can be used to show that the norms in these two $C^*$-algebras agree), as desired in the first item. The remainder isomorphisms in this Proposition follow from the first one.\n\\hfill{$\\Box$}\n\\par\nNow we recall the notion of strong Lipschitz homotopy \\cite{Yu97, Yu98, Yu00}.\n\\par\nLet $(Y_\\gamma)_{\\ggnn}$ and $(\\Delta_\\gamma)_{\\ggnn}$ be two families of uniformly bounded closed subsets of $\\pdxn$, $\\nn$,\nfor some $d\\geq 0$, such that $\\gamma\\in Y_\\gamma$, $\\gamma\\in \\Delta_\\gamma$ for all $\\gamma\\in \\Gamma_n, \\nn$.  A map\n$$g: \\; \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma \\longrightarrow \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} \\Delta_\\gamma $$\nis said to be {\\em Lipschitz} if\n\\\\ \\indent (1) $g(Y_\\gamma)\\subseteq \\Delta_\\gamma$ for each ${\\gamma\\in \\Gamma_n, \\nn}$;\n\\\\ \\indent (2) there exists a constant $c>0$, independent of $\\ggnn$, such that\n$$ d(g(x),g(y))\\leq c \\cdot d(x,y) $$\nfor all $x,y\\in Y_\\gamma$, $\\ggnn$.\n\\par\nLet $g_1,g_2$ be two Lipschitz maps from $\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma$ to $\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} \\Delta_\\gamma$.\nWe say $g_1$ is {\\it strongly Lipschitz homotopy} equivalent to $g_2$ if there exists a continuous map\n$$ F: \\; [0,1]\\times \\Big( \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma \\Big)\\longrightarrow \\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} \\Delta_\\gamma $$\nsuch that\n\\\\ \\indent (1) $F(0,x)=g_1(x)$, $F(1,x)=g_2(x)$ for all $x\\in\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma$;\n\\\\ \\indent (2) there exists a constant $c>0$ for which $ d(F(t,x),F(t,y))\\leq c \\cdot d(x,y)$ for all $x,y\\in Y_\\gamma$, $\\ggnn$, $t\\in [0,1]$;\n\\\\ \\indent (3) $F$ is equicontinuous in $t$, i.e., for any $\\varepsilon>0$ there exists $\\delta>0$ such that $d(F(t_1,x),F(t_2,x))<\\varepsilon $ for all\n$x\\in\\bigsqcup_{{\\gamma\\in \\Gamma_n, \\nn}} Y_\\gamma$ whenever $|t_1-t_2|<\\delta$.\n\\par\nWe say $(Y_\\gamma)_{\\ggnn}$ is {\\em strongly Lipschitz homotopy} equivalent to $(\\Delta_\\gamma)_{\\ggnn}$ if there exist Lipschitz maps\n$g_1:\\bigsqcup Y_\\gamma\\to \\bigsqcup \\Delta_\\gamma$ and $g_2:\\bigsqcup \\Delta_\\gamma\\to \\bigsqcup Y_\\gamma$ such that $g_1g_2$ and $g_2g_1$ are\nrespectively strongly Lipschitz homotopy equivalent to identity maps.\n\\par\nDefine $A^*_{L, 0, \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$ to be the $C^*$-subalgebra of\n$$A^*_{L,  \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$$\nconsisting of those functions $f$ such that $f(0)=0$.\n\\par\n{\\bf Lemma 6.12.} (cf. \\cite{Yu00})  {\\it\nIf $(Y_\\gamma)_{\\ggnn}$ is strongly Lipschitz homotopy equivalent to $(\\Delta_\\gamma)_{\\ggnn}$, then $K_*\\big(A^*_{L, 0, \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)\\big)$ is isomorphic to $K_*\\big(A^*_{L, 0, \\infty}\\big( (\\Delta_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)\\big)$.\n}\n\\hfill{$\\Box$}\n\\par\nLet $e$ be the evaluation homomorphism from $A^*_{L,  \\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$   to $A^*_{\\infty}\\big( (Y_\\gamma; \\, \\gamma\\in \\Gamma_n)_\\nn  \\big)$  given by\n$e(f)=f(0)$.\n\\par\n{\\bf Proposition 6.13.} (cf. \\cite{Yu00}) {\\it For any $d\\geq 0$, let $\\Delta_\\gamma$ be a simplex in $\\pdxn$ for all $\\ggnn$ with $\\gamma\\in \\Delta_\\gamma$. Then the evaluation map\n$$ e_*:\\, K_*\\big(A^*_{L, \\infty} \\big( (\\Delta_\\gamma; \\gamma\\in \\Gamma_n)_\\nn \\big)\\big) \\rightarrow  K_*\\big(A^*_{\\infty} \\big( (\\Delta_\\gamma; \\gamma\\in \\Gamma_n)_\\nn \\big)\\big) $$\nis an isomorphism.\n}\n\\par\n{\\bf Proof.} (cf.  \\cite{Yu00}) Note that $(\\Delta_\\gamma)_{\\ggnn}$ is strongly Lipschitz homotopy equivalent to $(\\{\\gamma\\})_{\\ggnn}$. By an argument of Eilenberg swindle,  we have\n$K_*\\big(A^*_{L,0, \\infty}\\big( (\\{\\gamma\\}; \\gamma\\in \\Gamma_n )_\\nn \\big)\\big)=0$. Consequently, Proposition 6.13 follows from Lemma 6.12 and the six-term exact sequence of $C^*$-algebra $K$-theory.\n\\hfill{$\\Box$}\n\\par\nWe are now ready to give a proof to Theorem 6.7.\n\\par\n{\\bf Proof of Theorem 6.7. }  By Proposition 6.11, we have the following commutative diagram\n\\[\n\\xymatrix{\n  \\displaystyle\\lim_{d\\to\\infty}    \\culmpdxna_O         \\ar[d]_{\\cong}          \\ar[r]^{\\quad e}\n            &    \\displaystyle\\lim_{d\\to\\infty}      \\cumpdxna_O          \\ar[d]^{\\cong}                   \\\\\n  \\displaystyle\\lim_{S\\to\\infty}     A^*_{L, \\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)        \\ar[r]^{\\quad e}\n            &    \\displaystyle\\lim_{s\\to\\infty}     A^*_{\\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)\n     }\n\\]\nwhich induces the following commutative diagram on $K$-theory\n\\[\n\\xymatrix{\n  \\displaystyle\\lim_{d\\to\\infty}    K_*\\Big(  \\culmpdxna_O  \\Big)       \\ar[d]_{\\cong}          \\ar[r]^{e_*}\n            &    \\displaystyle\\lim_{d\\to\\infty}    K_*\\Big(   \\cumpdxna_O   \\Big)       \\ar[d]^{\\cong}                   \\\\\n  \\displaystyle\\lim_{S\\to\\infty}  K_*\\Big(   A^*_{L, \\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)   \\Big)     \\ar[r]^{e_*}\n            &    \\displaystyle\\lim_{s\\to\\infty}    K_*\\Big(    A^*_{\\infty} \\big( (\\Delta_\\gamma(S); \\gamma\\in \\Gamma_n)_\\nn \\big)   \\Big)\n     }\n\\]\nNow Theorem 6.7 follows from Proposition 6.13.\n\\hfill{$\\Box$}\n\\subsection*{\\it \\S 6.3. Proof of Theorem 6.1}\nWe are now able to prove Theorem 6.1, the main result of this section, by gluing together the pieces we studied above.\n\\par\n{\\bf Proof of Theorem 6.1.}  For all $r>0$, define\n$$O^{(r)}_{n, x} \\;:=\\bigcup_{\\gamma\\in B_{X_n}(x, l_n)} B_{\\mathbb{R}_+\\times V_n}\\Big( t_x(\\gamma)(s(\\gamma)), \\, r   \\Big) $$\nfor all $x\\in X_n, \\nn$. Then\n$$O^{(r)}:=\\big( O^{(r)}_{n, x} \\big)_{x\\in X_n, \\nn}$$\nis a coherent system of open subsets. For any $d\\geq 0$, if $r<r'$ then $O^{(r)}_{n, x}\\subseteq O^{(r')}_{n, x}$ so that\n$$\\cumpdxna_{O^{(r)}} \\subseteq \\cumpdxna_{O^{(r')}}   \\;,$$\n$$\\culmpdxna_{O^{(r)}} \\subseteq \\culmpdxna_{O^{(r')}} \\;. $$\nBy the definition of these ideals, we have\n$$\\cumpdxna = \\lim_{r\\to \\infty} \\cumpdxna_{O^{(r)}}  \\;, $$\n$$\\culmpdxna= \\lim_{r\\to \\infty} \\culmpdxna_{O^{(r)}} \\;. $$\nConsequently,  it suffices to show that,  for each $r_0> 0$, the evaluation map\n\\[\n\\begin{array}{ll}\ne_*: &   \\displaystyle \\lim_{d\\to \\infty} K_*\\Big( \\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{O^{(r)}} \\Big)             \\\\\n&   \\longrightarrow  \\displaystyle \\lim_{d\\to \\infty} K_*\\Big(   \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\cumpdxna_{O^{(r)}} \\Big)\n\\end{array}\n\\]\nis an isomorphism.\n\\par\nLet $r_0>0$. Since $(X_n)_\\nn$ has uniform bounded geometry, there exists $N>0$ such that $\\# B(x, r_0)<N$ for all $x\\in X_n, \\nn$.\nIt follows that there exists an integer $J_{r_0}>0$ independent of $n$ such that we have the following decompositions for all $\\nn$:\n\\begin{itemize}\n\\item $X_n=\\bigcup_{j=1}^{J_{r_0}} \\Gamma_n^{(j)}$ for $J_{r_0}$ subspaces $\\Gamma_n^{(j)}$ of $X_n$;\n\\item $\\Gamma_n^{(j)}\\cap \\Gamma_n^{(j')}=\\emptyset$ whenever $j\\not= j'$;\n\\item for any $x\\in X_n$ and any distinct $\\gamma, \\gamma'\\in \\Gamma_n^{(j)}\\cap B_{X_n}(x, l_n)$,\n\t\t\t$$d\\Big( t_x(\\gamma)(s(\\gamma)), \\, t_x(\\gamma')(s(\\gamma'))  \\Big) >2r_0$$\n\t\tin $V_n$.\n\\end{itemize}\n\\par\nFor any $0<r<r_0$ and each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$, let\n$$O^{(r, j)}_{n,x} = \\bigcup_{\\gamma\\in \\Gamma_n^{(j)}\\cap B_{X_n}(x, l_n)}  B_{\\mathbb{R}_+\\times V_n}\\big( t_x(\\gamma)(s(\\gamma)), \\, r   \\big) $$\nfor all $x\\in X_n, \\nn$. Then the system\n$$O^{(r, j)}=\\Big( O^{(r, j)}_{n,x} \\Big)_{x\\in X_n, \\nn}$$\nis coherent for each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$, and\n$$O^{(r)}=\\bigcup_{j=1}^{J_{r_0}} O^{(r, j)}.$$\nFor each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$,  let\n$$\\Gamma^{(j)}=\\big( \\Gamma_n^{(j)} \\big)_\\nn.$$\nIt is clear that the system $O^{(r, j)}$ or\n$$O^{(r, j)} \\cap \\big( \\cup_{i=1}^{j-1} O^{(r, i)}  \\big)$$\nis $(\\Gamma^{(j)}, r)$-separate for each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$.\n\\par\nBy a construction of ``partition of unity'' as in the proof of Lemma 6.3 in \\cite{Yu00}, for each $j\\in \\{1, 2, \\cdots, J_{r_0}\\}$, we have\n$$\\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{O^{(r, j)}} +\\, \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$  =  \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\cumpdxna_{\\cup_{i=1}^{j} O^{(r, i)}} \\;\\; ,$$\n$$ \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{O^{(r, j)}}\\cap \\, \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$ = \\displaystyle \\lim_{r<r_0, r\\to r_0} \\cumpdxna_{O^{(r, j)} \\cap \\big( \\cup_{i=1}^{j-1} O^{(r, i)}  \\big)} \\;\\; ,$$\n$$\\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{O^{(r, j)}}+\\, \\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$ = \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\culmpdxna_{\\cup_{i=1}^{j} O^{(r, i)}} \\;\\; ,$$\n$$\\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{O^{(r, j)}}\\cap \\,\\displaystyle \\lim_{r<r_0, r\\to r_0} \\culmpdxna_{\\cup_{i=1}^{j-1} O^{(r, i)}} $$\n$$  = \\displaystyle \\lim_{r<r_0, r\\to r_0}  \\culmpdxna_{O^{(r, j)} \\cap \\big( \\cup_{i=1}^{j-1} O^{(r, i)}  \\big)} \\;\\; .$$\n\\par\nNow, Theorem 6.1 follows from Theorem 6.7, together with a Mayer-Vietoris sequence argument (cf. Sect. 3 of \\cite{HRY}).\n\\hfill{$\\Box$}\n\n\n\n\\section{Geometric analogue of Bott periodicity}\nIn this section, we shall define maps $\\alpha$, $\\beta$, $\\alpha_L$, $\\beta_L$ to build the following commutative diagram\n\\[\n\\xymatrix{\nK_*\\Big( \\culmpdxn \\Big) \\ar[r]^{\\quad e_*} \\ar[d]^{(\\beta_L)_*} & K_*\\Big( \\cumpdxn \\Big) \\ar[d]^{\\beta_*} \\\\\nK_*\\Big( \\culmpdxna \\Big)  \\ar@<1ex>[u]^{(\\alpha_L)_*} \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxna \\Big )\\;. \\ar@<1ex>[u]^{\\alpha_*}\n}\n\\]\nand show a geometric analogue of the Bott periodicity.\n\\par\nThe maps $\\alpha$, $\\beta$, $\\alpha_L$, $\\beta_L$ will be constructed as {\\em asymptotic morphisms}. The reader is referred to\n\\cite{GHT, HK97, HK01, HKT, HG, Yu00} for backgrounds on asymptotic morphisms and the sources of most of the ideas behind the current section. In what follows, we shall use the graded formalism from \\cite{HG}.\n\n\\subsection*{\\it \\S 7.1. The Bott maps $\\beta$ and $\\beta_L$}\nIn this subsection, we define the Bott maps  $\\beta$ and $\\beta_L$.\nRecall that $(X_n)_\\nn$ is a sequence of finite metric spaces with uniform bounded geometry which admit a fibred coarse embedding into Hilbert space as in Definition 5.4.\nFor each $d\\geq 0$, $\\zdn$ is a countable dense subset of $\\pdxn$. And\n$$V_n:= \\mbox{ affine-span }\\Big\\{ t_x(z)(s(z)) \\, \\Big| \\, z\\in B(x, l_n), x\\in X_n \\Big\\}$$\nis a finite dimensional affine subspace of $H$ for each $\\nn$.\n\\par\nBy Definition 5.1,  for each $x\\in \\zdn$, the inclusion of the $0$-dimensional affine subspace $\\{t_x(x)(s(x))\\}$ into $V_n$ induces a $*$-homomorphism\n$$\\beta(x): \\, \\cS\\cong \\cA\\big( \\{t_x(x)(s(x))\\} \\big)  \\rightarrow \\cA(V_n)$$\nby the formula\n$$\\Big( \\beta(x) \\Big) (g)= g \\big(   X\\widehat\\otimes 1+ 1\\widehat\\otimes C_{V_n,t_x(x)(s(x))}   \\big)  \\,,$$\nwhere\n$$C_{V_n,t_x(x)(s(x))}:\\, V_n\\rightarrow V_n^0 \\subseteq \\Cliff(V_n^0)$$\nis the function $v\\mapsto v- t_x(x)(s(x))\\in V_n^0 \\subseteq \\Cliff(V_n^0)$ for all $v\\in V_n$.\n\\par\n{\\bf Definition 7.1.} Let $d\\geq 0$. For each $t\\in [1, \\infty)$, define a map\n$$\\beta_t:\\, \\cS\\widehat\\otimes \\cupdxn \\longrightarrow \\cumpdxna$$\nfor $g\\in \\cS$, $\\ttt\\in \\cupdxn$ by the formula\n$$\\beta_t\\big( g\\widehat\\otimes T \\big)=\\Big[ \\, \\Big(      \\big(  \\beta_t\\big( g\\widehat\\otimes T \\big) \\big)^{(0)}, \\cdots, \\big(  \\beta_t\\big( g\\widehat\\otimes T \\big) \\big)^{(n)}, \\cdots \\Big)\\,\\Big] \\,,$$\nwhere\n$$\\big(  \\beta_t\\big( g\\widehat\\otimes T \\big) \\big)^{(n)}(x, y)=\\big( \\beta(x) \\big) (g_t) \\widehat\\otimes T^{(n)}(x, y)$$\nfor $x, y\\in \\zdn$, $\\nn$, and $g_t(r)=g(\\frac{r}{t})$ for all $r\\in \\mathbb{R}$.\n\\hfill{$\\Box$}\n\\par\n{\\bf Definition 7.2.}  Let $d\\geq 0$. For each $t\\in [1, \\infty)$, define a map\n$$(\\beta_L)_t:\\, \\cS\\widehat\\otimes \\culpdxn \\longrightarrow \\culmpdxna$$\nby the formula\n$$\\Big(  (\\beta_L)_t (f) \\Big) (\\tau)= \\beta_t\\big(  f(\\tau)  \\big)$$\nfor $\\tau\\in \\mathbb{R}_+=[0, \\infty)$.\n\\hfill{$\\Box$}\n\\par\n{\\bf Lemma 7.3.} {\\it\nFor each $d\\geq 0$, the maps $(\\beta_t)_{t\\geq 1}$ and $\\big((\\beta_L)_t\\big)_{t\\geq 1}$ extend to asymptotic morphisms\n$$\\beta:\\; \\cS\\widehat\\otimes \\cumpdxn  \\rightarrow \\cumpdxna   \\,,$$\n$$\\beta_L:\\; \\cS\\widehat\\otimes \\culmpdxn  \\rightarrow \\culmpdxna  \\,.$$\n}\n\\par\n{\\bf Proof.} For any $x, y\\in \\zdn$ with $n\\geq N$ for some $N>0$ large enough such that both  $x$ and $y$ are in the regions of trivializations of $t_x$ and $t_y$, define\n$1$-dimensional affine subspaces of $V_n$:\n$$W^x:=W_{\\{x, y\\}}(x):=\\mbox{ affine-span } \\Big\\{ t_x(x)(s(x)),\\;  t_x(y)(s(y)) \\Big\\} \\,,$$\n$$W^y:=W_{\\{x, y\\}}(y):=\\mbox{ affine-span } \\Big\\{ t_y(x)(s(x)), \\; t_y(y)(s(y)) \\Big\\} \\,.$$\nThen, by Definition 5.1 on $\\beta_{ba}$, we have\n$$\\beta(x)=\\beta_{V_n, W^x}\\circ \\beta_{W^x, \\{ t_x(x)(s(x)) \\}} \\,,$$\n$$\\beta(y)=\\beta_{V_n, W^y}\\circ \\beta_{W^y, \\{ t_y(y)(s(y)) \\}} \\,.$$\nMoreover, we have  $W^x=t_{xy}\\big( W^y \\big)$ and\n$$\\beta_{W^x, \\{ t_x(y)(s(y)) \\}}(g)=\\big( t_{xy} \\big)_* \\Big(  \\beta_{W^y, \\{ t_y(y)(s(y)) \\}}(g)  \\Big)$$\nfor all $g\\in \\cS$, where recall that $t_{xy}=t_x\\circ t_y^{-1}: H\\to H$ and\n$$\\big( t_{xy} \\big)_*: \\;  \\cA(W_C(y)) \\rightarrow \\cA(W_C(x))$$\nmapping $g\\widehat\\otimes h$ to $g\\widehat\\otimes (t_{xy})_*(h)$, is defined in Definition 5.3 for all $C\\subseteq B_{X_n}(x, l_n)\\cap B_{X_n}(y, l_n)$.\n\\par\nFor the generators $g(x)=\\frac{1}{x\\pm i}$ of $\\cS=C_0(\\mathbb{R})$, it is standard argument to verify, for $t\\in [1, \\infty)$, that\n\\[\n\\begin{array}{ll}\n  &  \\Big\\| \\beta_{W^x, \\{ t_x(x)(s(x)) \\}}(g_t) - \\big( t_{xy} \\big)_* \\Big(  \\beta_{W^y, \\{ t_y(y)(s(y)) \\}}(g_t)    \\Big)  \\Big\\|              \\\\\n= &  \\big\\| \\beta_{W^x, \\{ t_x(x)(s(x)) \\}}(g_t) -   \\beta_{W^x, \\{ t_x(y)(s(y)) \\}}(g_t)      \\big\\|              \\\\\n\\leq & \\frac{1}{t} \\|  t_x(x)(s(x)) - t_x(y)(s(y)) \\|     \\\\\n\\leq & \\frac{1}{t} \\rho_2(d(x, y)) \\; \\longrightarrow 0 \\;\\; (t\\to \\infty)  \\,.\n\\end{array}\n\\]\nIt follows from an approximation argument, together with \\cite{Yu00}(Lemma 7.6) and \\cite{HG}(Lemma 3.2), that for all $d\\geq 0$, $R>0, r>0, c>0, \\varepsilon>0$, there exists\n$t_0>1$ such that for all $x, y\\in \\zdn, \\nn, $ with $d(x, y)\\leq R$, for all $t\\geq t_0$ and all $g\\in \\cS$ with\n$$\\supp(g)\\subseteq [-r, r], \\quad\\quad\\quad  \\|g'\\|_\\infty\\leq c  \\,,$$\nwe have\n$$\\Big\\| \\big(  \\beta(x) \\big) (g_t) -\\big( t_{xy} \\big)_*\\Big(   \\big(  \\beta(y) \\big) (g_t)  \\Big)   \\Big\\| < \\varepsilon.$$\n($(t_{xy})_*$ is as in Definition 5.5) As a result, the maps $\\beta_t$ ($t\\geq 1$) define a $*$-homomorphism $\\beta$ from $\\cS\\widehat\\otimes \\cupdxn$ to the asymptotic $C^*$-algebra\n$$\\mathfrak{A}\\Big(  \\cumpdxna  \\Big) := \\frac\n{C_b\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxna   \\Big)}\n{C_0\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxna   \\Big)}\n$$\nsatisfying $\\|\\beta(g\\widehat\\otimes T)\\|\\leq \\|g\\|\\cdot \\|T\\|$ for all $g\\in \\cS$ and $T\\in \\cupdxn$. Hence, by the universality of the max norm and the max tensor product,\n$\\beta$ extends to a $C^*$-homomorphism from\n$$\\cS \\widehat\\otimes _{\\max} \\cumpdxn $$\nto $\\mathfrak{A}\\big(  \\cumpdxna  \\big)$. Since $\\cS$ is nuclear, we conclude that $\\beta_t$ extends to an asymptotic morphism from $\\cS \\widehat\\otimes \\cumpdxn$ to\n$\\cumpdxna$. The case for $\\beta_L$ is similar.\n\\hfill{$\\Box$}\n\n\n\n\\subsection*{\\it \\S 7.2. The Dirac maps $\\alpha$ and $\\alpha_L$      }\nIn this subsection, we define the Dirac maps $\\alpha$ and $\\alpha_L$. To begin with, we recall the definition of  the Bott-Dirac operator on a finite dimensional\naffine subspace of the Hilbert space $H$.\n\\par\nLet $V$ be a finite dimensional affine subspace of $H$, and let $V^0$ be the corresponding finite dimensional linear space consisting of differences of elements of $V$. Let\n$$L^2(V):=L^2(V, \\Cliff(V^0))$$\nbe the graded infinite dimensional complex Hilbert space of square integrable $\\Cliff(V^0)$-valued functions on $V$, where $V$ is endowed with the Lebesgue measure induced from the inner\nproduct on $V^0\\subseteq H$. The grading of $L^2(V)$ is inherited from the Clifford algebra $\\Cliff(V^0)$, which is also considered as a graded Hilbert space in such a way that\nthe monomials $e_{i_1}\\cdots e_{i_p}$ associated to an orthonormal basis of $V^0$ are orthonormal.\n\\par\nThe {\\em  Dirac operator $D_V$ of $V$}  is the unbounded operator on $L^2(V)$ defined by the formula\n$$\\big( D_V\\xi \\big) (v) =\\sum_{i=1}^n (-1)^{\\mbox{degree} (\\xi)} \\frac{\\partial \\xi}{\\partial x_i} (v) \\cdot e_i$$\nwhere $\\{e_1, \\cdots, e_n\\}$ is an orthonormal basis for $V^0$, $\\xi\\in L^2(V)$, $v\\in V$, and $\\{x_1, \\cdots, x_n\\}$ is the coordinates dual to  $\\{e_1, \\cdots, e_n\\}$.\nThe domain of $D_V$ is the Schwartz subspace of $L^2(V)$. Note that $D_V$ does not depend on the choice of an orthonormal basis for $V^0$ (and does not depend on a basis point in $V$).\n\\par\nThe {\\em Clifford operator $C_{V, v_0}$ of $V$ at $v_0\\in V$} is an unbounded operator on $L^2(V)$ defined by the formula\n$$\\big( C_{V, v_0} \\xi \\big) (v) = (v-v_0) \\cdot \\xi(v)$$\nfor all $v\\in V$, $\\xi\\in L^2(V)$, where the multiplication $\\cdot$ is the Clifford multiplication of $v-v_0\\in V^0\\subset \\Cliff(V^0)$ and $\\xi(v)\\in \\Cliff(V^0)$.\nThe domain of $C_{V,v_0}$ is also the Schwartz subspace of $L^2(V)$. Note that the Clifford operator  $C_{V, v_0}$  depends on  the choice of the base point $v_0$.\nIf $V$ is actually a linear subspace, and $v_0=0\\in V$, then we also denote $C_V:=C_{V, 0}$.\n\\par\nThe {\\em  Bott-Dirac operator of  $V$ at $v_0\\in V$} is\n$$B_{V, v_0} =D_V+C_{V, v_0}$$\nwith domain again the Schwartz subspace of $L^2(V)$. Denote by $\\xi_{V, v_0}$ the unit vector of $L^2(V)=L^2(V, \\Cliff(V^0))$:\n$$\\xi_{V, v_0}(v)= \\pi^{-\\frac{n}{4}}\\cdot e^{-\\frac{\\|v-v_0\\|^2}{2}} \\; ,$$\n where $n=\\dim(V^0)$. Then $\\xi_{V, v_0}$ spans the kernel of $B_{V, v_0}$, which is a $1$-dimensional subspace of $L^2(V)$.\nIf $V$ is a linear subspace, and $v_0=0\\in V$, then we also denote $B_V:=B_{V, 0}$.\n\\par\nLet $V_a, V_b$ be finite dimensional affine subspaces of $H$ such that $V_a\\subseteq V_b$ and $V_b=V_{ba}^0\\oplus V_a$. Then\n$$L^2(V_b)\\cong L^2(V_{ba}^0)\\widehat\\otimes L^2(V_a)$$\nand $L^2(V_a)$ is regarded as a subspace of $L^2(V_b)$ via the isometry from $L^2(V_a)$ to $L^2(V_b)$ given by\n$$\\xi \\mapsto \\xi_{V_{ba}^0, 0}\\widehat\\otimes \\xi.$$\n\\par\nRecall also that an affine isometry $t: V\\to V$ canonically induces a unitary on $L^2(V)$ and moreover, a $*$-isomorphism\n$$t_*: \\cK(L^2(V))\\rightarrow \\cK(L^2(V))$$\nby conjugation with the unitary.\n\\par\n{\\bf We now come back to the case of interest.} Let $(X_n)_\\nn$ be a sequence of finite metric spaces with uniform bounded geometry which admit a fibred coarse embedding\ninto Hilbert space as described in Definition 5.4. Recall in Definition 5.5, for all $\\nn$,\n$$V_n=\\mbox{ affine-span }\\big\\{ t_x(z)(s(z)) \\,|\\, z\\in B_{X_n}(x, l_n), x\\in X_n \\big\\}.$$\nDefine\n$$E_n:= \\mbox{ linear-span } \\Big\\{ V_n,\\, 0 \\Big\\} \\subseteq H  \\,,$$\nwhich is a  finite dimensional {\\em linear} subspace of $H$ containing $V_n$.\nDenote\n$$L^2_n:=L^2(E_n, \\Cliff(E_n))$$\nand denote by $\\cK(L^2_n)$ the graded $C^*$-algebra of all compact operators on the graded Hilbert space $L^2_n$ for all $\\nn$.\n\\par\n{\\bf Definition 7.4.} For all $x, z\\in X_n (\\nn) $ with $B(x, l_n)\\cap B(z, l_n)\\not=\\emptyset$, we define an isomorphism\n$$(t_{xz})_*:\\, \\cK(L^2_n)\\widehat\\otimes \\cK \\rightarrow \\cK(L^2_n)\\widehat\\otimes \\cK$$\nas follows. Denote\n$$W_x:=\\mbox{ affine-span }\\big\\{ t_x(w)(s(w)) \\,\\big| \\, w\\in  B(x, l_n)\\cap B(z, l_n)  \\,  \\}, $$\n$$W_z:=\\mbox{ affine-span }\\big\\{ t_z(w)(s(w)) \\,\\big| \\, w\\in  B(x, l_n)\\cap B(z, l_n)  \\,  \\}. $$\nThen $W_x=t_{xz}(W_z)$. Denote $W_x^\\perp=E_n\\ominus W_x$ and $W_z^\\perp=E_n\\ominus W_z$ the linear orthogonal complements of $W_x, W_y$ in $E_n$. Choose a unitary operator\n$U_{xz}: \\, W_z^\\perp \\rightarrow W_x^\\perp $. Then\n$$U_{xz}\\oplus t_{xz}: \\, E_n=W_z^\\perp\\oplus W_z \\longrightarrow E_n=W_x^\\perp\\oplus W_x$$\nis an affine isometry from $E_n$ onto $E_n$. We define\n$$(t_{xz})_*:=(U_{xz}\\oplus t_{xz})_*\\widehat\\otimes 1: \\; \\cK(L^2_n)\\widehat\\otimes \\cK \\rightarrow \\cK(L^2_n)\\widehat\\otimes \\cK .$$\nMoreover, in Rips complexes for each $d\\geq 0$, if $x, z\\in \\zdn\\subseteq \\pdxn, \\nn$, with $x\\in \\Star(\\bar x)$ and $z\\in \\Star(\\bar z)$, we define $(t_{xz})_*=(t_{\\bar{x}\\bar{z}})_*$.\n\\par\n{\\bf Remark 7.5.} Let  $\\xi_{W_x^\\perp}$ and $\\xi_{W_z^\\perp}$ be the unit vectors in the kernels of the Bott-Dirac operators at the origin of the linear spaces $L^2(W_x^\\perp)$ and $L^2(W_z^\\perp)$.\nThen  $\\xi_{W_x^\\perp} = U_{xz}\\big( \\xi_{W_z^\\perp} \\big)$. This fact implies in the sequel actual applications,  $(t_{xz})_*$ does not depend on the choice of the unitary $U_{xz}$.\n\\par\n{\\bf Definition 7.6.} For each $d\\geq 0$, define $\\cupdxnk$ to be the set of all equivalence classes $\\ttt$ of sequences\n\t$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots)$$\ndescribed as follows:\n\\begin{itemize}\n\\item[(1)] $T^{(n)}$ is a bounded function from $\\zdn \\times \\zdn$ to $\\cK(L^2_n)\\widehat\\otimes\\cK$ for all $\\nn$;\n\\item[(2)] for any bounded subset $B\\subset \\pdxn$, the set\n\t\t\t$$\\{ (x, y)\\in B\\times B \\cap \\zdn\\times \\zdn \\; | \\; T^{(n)} (x, y)\\not= 0 \\}$$\n\t\t\tis finite;\n\\item[(3)] there exists $L>0$ such that\n\t\t\t$$\\#\\{y\\in \\zdn \\; | \\; T^{(n)}(x, y)\\not= 0 \\}<L, \\quad \\#\\{y\\in \\zdn \\; | \\; T^{(n)}(y, x)\\not= 0 \\}<L $$\n\t\t\tfor all $x\\in \\zdn, \\nn$;\n\\item[(4)] there exists $R>0$ such that $T^{(n)}(x, y)=0$ whenever $d(x, y)>R$ for $x, y\\in \\zdn, \\; \\nn$.\n\\end{itemize}\nThe equivalence relation $\\sim$ on these sequences is defined by\n$$(T^{(0)}, \\cdots, T^{(n)}, \\cdots) \\sim (S^{(0)}, \\cdots, S^{(n)}, \\cdots)$$ if and only if\n$$\\lim_{n\\to \\infty} \\sup_{x, y\\in \\zdn} \\big\\| T^{(n)}(x, y) - S^{(n)}(x, y) \\big\\|_\\cK =0.$$\nThe product structure for $\\cupdxnk$ is defined as follows. For any two elements $\\ttt$ and $S=\\big[ (S^{(0)}, \\cdots, S^{(n)}, \\cdots) \\big]$ in $\\cupdxnk$, their product is defined to be\n$$\\big[ \\big( (TS)^{(0)}, \\cdots, (TS)^{(n)}, \\cdots \\big) \\big], $$\nwhere there exists a sufficiently large $N\\in \\mathbb{N}$ depending on the propagation of the two representative sequences,\nsuch that $ (TS)^{(n)}=0$ for all $n=0, 1, 2, \\cdots, N-1$ and\n$$(TS)^{(n)}(x, y)=\\sum_{z\\in \\zdn} \\Big( T^{(n)}(x, z) \\Big) \\cdot \\Big( \\big(t_{xz}\\big)_* \\big( S^{(n)}(z, y) \\big) \\Big)$$\nfor all $x, y\\in \\zdn, \\; n\\geq N$, where $\\big(t_{xz}\\big)_*$ is defined as in Definition 7.4.\n\\par\nThe $*$-structure for $\\cupdxnk$ is defined by the formula\n$$ \\big[ \\big( T^{(0)}, \\cdots, T^{(n)}, \\cdots \\big) \\big]^*=   \\big[ \\big( (T^*)^{(0)}, \\cdots, (T^*)^{(n)}, \\cdots \\big) \\big] \\,,$$\nwhere\n$$(T^*)^{(n)}(x, y)=\\big( t_{xy} \\big)_* \\Big( \\big( T^{(n)}(y, x)\\big)^*\\Big)$$\nfor all but finitely many $n$, and $0$ otherwise.\n\\par\nNow, $\\cupdxnk$ is made into a $*$-algebra by using the additional usual matrix operations. Define $\\cumpdxnk$ to be the completion of $\\cupdxnk$ with respect to the norm\n\\begin{footnotesize}\n$$\\big\\|T\\big\\|_{\\max} :=\\sup\\Big\\{ \\big\\|\\phi(T)\\big\\|_{\\cB(H_\\phi)} \\; \\Big | \\; \\phi:\\cupdxnk \\to \\cB(H_\\phi), \\mbox{ a $*$-representation} \\Big\\}. $$\n\\end{footnotesize}\n\\hfill{$\\Box$}\n\\par\nSimilarly we define the localization algebra $\\culmpdxnk$.\n\\par\n{\\bf Definition 7.7.} Let $d\\geq 0$. For each $t\\in [1, \\infty)$ define a map\n$$\\alpha_t:\\; \\cupdxna\\rightarrow \\cumpdxnk$$\nby the formula\n$$\\alpha_t(T)=\\Big[ \\Big(  \\big(\\alpha_t(T)\\big)^{(0)}, \\cdots,  \\big(\\alpha_t(T)\\big)^{(n)}, \\cdots     \\Big)  \\Big]$$\nfor $\\ttt\\in \\cupdxna$, with\n$$\\big(\\alpha_t(T)\\big)^{(n)}(x, y)=\\Big( \\theta_t^k(x) \\Big) \\big( T_{1}^{(n)}(x, y)\\big)$$\nfor all $x, y\\in \\zdn, \\nn$, where\n\\begin{itemize}\n\\item[(1)] there exists $k\\geq 0$ independent of $\\nn$ as in condition (6) of Definition 5.5 such that\n\t\t\t\t\t\t$$ T^{(n)}(x, y) = \\Big( \\beta_{V_n, W_k(x)} \\widehat\\otimes 1 \\Big) \\big( T^{(n)}_1 (x, y) \\big)$$\n\t\tfor some $T_1^{(n)}(x, y)\\in \\cA(W_k(x))\\widehat\\otimes \\cK$ of the form $\\sum_{i=1}^K g_i\\widehat\\otimes h_i\\widehat\\otimes k_i$ where $g_i\\in \\cS$, $h_i\\in \\cC(W_k(x))$,\n\t\t\t$k_i\\in \\cK$ for $1 \\leq i \\leq K$.\n\\item[(2)] The map\n\t\t$$ \\theta_t^k(x):\\, \\cA(W_k(x))\\widehat\\otimes \\cK \\rightarrow \\cK(L^2_n)\\widehat\\otimes \\cK$$\n\t\tis defined  by the formula\n\t\t$$\\Big( \\theta_t^k(x) \\Big) \\big( g\\widehat\\otimes h \\widehat\\otimes k \\big)\n\t\t\t= g_t\\big( B_{E_n\\ominus W_k(x)} \\widehat\\otimes 1 + 1\\widehat\\otimes D_{W_k(x)}  \\big)\\big( 1\\widehat\\otimes M_{h_t} \\big) \\widehat\\otimes k$$\n\t\tfor all $g\\in \\cS$, $h\\in \\cC(W_k(x))$, $k\\in \\cK$, $t\\geq 1$, $x\\in \\zdn, \\nn$, where\n\t\\begin{itemize}\n\t\\item[$\\bullet$]  $g_t(s)=g(s\/t)$ for $s\\in \\mathbb{R}$.\t\n\t\\item[$\\bullet$] $D_{W_k(x)}$ is the Dirac operator of the affine space\n\t\t\t\t\t$$W_k(x)=\\mbox{ affine-span } \\big\\{ t_x(z)(s(z)) \\, \\big| \\, z\\in B_{\\pdxn}(x, k) \\big\\} \\subset E_n .$$\n\t\\item[$\\bullet$] $B_{E_n\\ominus W_k(x)}$ is the Bott-Dirac operator at the origin $0$ of the linear orthogonal complement of $W_k(x)$ in $E_n$.\n\t\\item[$\\bullet$] for any $h\\in \\cC(W_k(x))$ and $t\\geq 1$, the function $h_t\\in \\cC(W_k(x))$ is defined by\n\t\t\t\t\t$$ h_t(v)= h\\Big( t_x(x)(s(x))+\\frac{1}{t} \\big( v-t_x(x)(s(x))   \\big)  \\Big)$$\n\t\t\t\tfor all $v\\in W_k(x)$.\n\t\\item[$\\bullet$] $M_{h_t}$ is the pointwise multiplication operator on $L^2(W_k(x), \\Cliff(W_k(x)^0))$\n\t\t\t\t\t$$\\Big( M_{h_t} \\xi \\Big) (v) = h_t(v)\\cdot \\xi(v)$$\n\t\t\t\tfor all $\\xi\\in L^2(W_k(x))$ and $v\\in W_k(x)$.\n\t\\item[$\\bullet$] $1\\widehat\\otimes M_{h_t}:\\, L^2_n\\to L^2_n$ is defined according to the tensor decomposition\n\t\t\t\t\t$$L^2_n\\cong L^2(E_n\\ominus W_k(x)) \\widehat\\otimes L^2(W_k(x)).$$\n\t\\end{itemize}\n\\end{itemize}\n\\par\n{\\bf Remark 7.8.} (1) The fact that $\\Big( \\theta_t^k(x) \\Big) \\big( g\\widehat\\otimes h \\widehat\\otimes k \\big) $ is in $\\cK(L^2_n)\\widehat\\otimes \\cK$ follows from the ellipticity of the Dirac operator\nand the Rellich lemma, see (\\cite{HKT} Definition 8). (2) By the proof of Proposition 4.2 in \\cite{HKT}, we know that $\\alpha_t(T)$ does not asymptotically depend on the choice of $k$.\n\\par\n{\\bf Definition 7.9.} For each $d\\geq 0$ and $t\\geq 1$, define\n$$(\\alpha_L)_t:\\; \\culpdxna\\rightarrow \\culmpdxnk$$\nby the formula\n$$\\Big( (\\alpha_L)_t(f) \\Big)(\\tau)=\\alpha_t\\big(f(\\tau) \\big)$$\nfor all $\\tau\\in [0, \\infty)$.\n\\par\n{\\bf Lemma 7.10.} For each $d\\geq 0$, the maps $(\\alpha_t)_{t\\geq 1}$ and $((\\alpha_L)_t)_{t\\geq 1}$ extend to asymptotic morphisms\n$$\\alpha:\\; \\cumpdxna\\rightarrow \\cumpdxnk \\,,$$\n$$\\alpha_L:\\; \\culmpdxna\\rightarrow \\culmpdxnk  \\,.$$\n\\par\n{\\bf Proof.} We first fix some notations. For a given large $\\ell \\geq 0$, and for any subset\n$$C\\subseteq B_{\\pdxn}(x, \\ell)\\cap B_{\\pdxn}(z, \\ell)$$\nwhere $x, z\\in \\zdn$  (for all but finitely many $n$)  with $d(x, z)<\\ell$ , denote\n$$W_C(x)=\\mbox{ affine-span }\\big\\{ t_x(w)(s(w)) \\, \\big| \\, w\\in C \\big\\},$$\n$$W_C(z)=\\mbox{ affine-span }\\big\\{ t_z(w)(s(w)) \\, \\big| \\, w\\in C \\big\\}.$$\nThen $W_C(x)=t_{xz}(W_C(z))$. For any affine subspace $W$ with $W_C(x)\\subseteq W\\subseteq E_n$, we identify $\\cA(W_C(x))$ with a subalgebra of $\\cA(W)$ via the map\n$\\beta_{W, W_C(x)}$ defined in Definition 5.1.\n\\par\n{\\it Step 1.}  For any $K>0$, $r>0$, $c>0$ and $x\\in \\zdn$, denote by\n\t\t$$\\Big[  \\cA(W_C(x))\\widehat\\otimes \\cK \\Big]_{K, r, c}$$\nthe subset of $\\cA(W_C(x))\\widehat\\otimes \\cK$ consisting of those elements of the form $\\sum_{i=1}^K g_i\\widehat\\otimes h_i\\widehat\\otimes k_i$ where $g_i\\in \\cS$, $h_i\\in \\cC(W_C(x))$,\n$k_i\\in \\cK$ such that (1) each $g_i$ is supported in $[-r, r]$; (2) each $g_i$ and $h_i$ is continuously differentiable and their derivatives satisfy $\\|g'_i\\|_\\infty\\leq c$ and\n $\\|\\nabla_v h_i\\|\\leq c$  for all $v\\in W_C(x)$ such that $\\|v-t_x(x)(s(x))\\|\\leq 1$.\n\\par\nIt follows from (\\cite{Yu00} Lemma 7.5) and (\\cite{HKT} Lemma 2.9) that for any $\\varepsilon>0$ there exists $t_0>1$ such that for all $t\\geq t_0$ and all\n$a, b\\in \\Big[  \\cA(W_C(x))\\widehat\\otimes \\cK \\Big]_{K, r, c} $, we have\n$$\\big\\| \\theta_t^\\ell(x)(ab)-\\theta_t^\\ell(x)(a)\\theta_t^\\ell(x)(b) \\big\\|<\\varepsilon, \\quad\\quad   \\big\\| \\theta_t^\\ell(x)(a^*)-\\theta_t^\\ell(x)(a)^* \\big\\|<\\varepsilon. $$\n\\par\n{\\it Step 2.} The affine isometry $t_{xz}: W_C(z)\\to W_C(x)$ induces a diagram\n\\[\n\\xymatrix{\n\\cA\\big(W_C(z)\\big)\\widehat\\otimes \\cK  \\ar[r]^{\\quad \\theta_t^\\ell(z)} \\ar[d]_{(t_{xz})_*}        &    \\cK(L^2_n)\\widehat\\otimes \\cK  \\ar[d]^{(t_{xz})_*}             \\\\\n\\cA\\big(W_C(x)\\big)\\widehat\\otimes \\cK  \\ar[r]^{\\quad \\theta_t^\\ell(x)}   &  \\cK(L^2_n)\\widehat\\otimes \\cK\n}\n\\]\n\\par\nIt follows from the proof of (\\cite{HKT} Proposition 4.2) and (\\cite{Yu00} Lemma 7.2) that the diagram is asymptotically commutative in the sense that, for any\n$R>0$ (relating propagations in $\\cupdxna$) and any large $\\ell$ (greater than $R+k$ where $k$ is as  Definition 5.5 (6)), for any  $K>0$, $r>0$, $c>0$ as above, and for any\n$\\varepsilon>0$, there exists $t_0>1$ such that for all $t\\geq t_0$ and all $b\\in \\Big[  \\cA(W_C(z))\\widehat\\otimes \\cK \\Big]_{K, r, c} $ , we have\n$$\\Big\\|   \\Big(\\theta_t^\\ell(x)\\Big) \\big( (t_{xz})_*(b) \\big) -  (t_{xz})_* \\Big(  \\big(\\theta_t^\\ell(z)\\big)(b)      \\Big) \\Big\\|<\\varepsilon,$$\nwhenever  $x, z\\in \\zdn$ satisfying  $d(x, z)\\leq R$.\n\\par\n{\\it Step 3.} From the above facts it follows that the maps $(\\alpha_t)_{t\\geq 1}$ define a $*$-homomorphism  $\\alpha$ from $\\cupdxna$ to the asymptotic $C^*$-algebra\n$$\\mathfrak{A}\\Big(  \\cumpdxnk  \\Big) := \\frac\n{C_b\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxnk   \\Big)}\n{C_0\\Big( \\big[ 1, \\infty \\big), \\; \\cumpdxnk   \\Big)}\n$$\nBy the universality of the max norm, $\\alpha$ extends to a $*$-homomorphism from the $C^*$-algebra $\\cumpdxna$ to $\\mathfrak{A}$. The case for $\\alpha_L$ is similar.\n\\hfill{$\\Box$}\n\n\\subsection*{\\it \\S 7.3. A geometric analogue of the Bott periodicity in finite dimension}\nNote that the asymptotic morphisms $\\beta$, $\\beta_L$, $\\alpha$, $\\alpha_L$  induce the following commutative diagram on $K$-theory:\n\\[\n\\xymatrix{\nK_*\\Big( \\culmpdxn \\Big) \\ar[r]^{\\quad e_*} \\ar[d]^{(\\beta_L)_*} & K_*\\Big( \\cumpdxn \\Big) \\ar[d]^{\\beta_*} \\\\\nK_*\\Big( \\culmpdxna \\Big)  \\ar[d]^{(\\alpha_L)_*} \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxna \\Big )\\;. \\ar[d]^{\\alpha_*}  \\\\\nK_*\\Big( \\culmpdxnk \\Big)  \\ar[r]^{\\quad e_*}  & K_*\\Big( \\cumpdxnk \\Big )\n}\n\\]\n\\par\nIn this subsection, we shall prove the following result, which is a geometric analogue of the Bott periodicity in finite dimension. The proof is adapted from (\\cite{Yu00} Proposition 7.7).\n\\par\n{\\bf Theorem 7.11.} {\\it\nFor each $d\\geq 0$, the compositions $\\alpha_*\\circ \\beta_*$ and $(\\alpha_L)_*\\circ (\\beta_L)_*$ are the identity.\n}\n\\par\n{\\bf Proof.} {\\it Step 1.} Let $\\gamma$ be the asymptotic morphism\n$$\\gamma:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk$$\ndefined by the formula\n$$\\gamma_t(g\\widehat\\otimes T)= \\Big[ \\, \\Big(    \\big(  \\gamma_t(g\\widehat\\otimes T)   \\big)^{(0)}, \\cdots, \\big(  \\gamma_t(g\\widehat\\otimes T)   \\big)^{(n)}, \\cdots     \\Big) \\Big]$$\nfor all $g\\in \\cS$ and $\\ttt\\in \\cupdxn$, where\n$$\\Big(  \\gamma_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y) = g_{t^2}\\big(  B_{E_n, t_x(x)(s(x))}  \\big)\\widehat\\otimes T^{(n)}(x, y)$$\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$, and $B_{E_n, t_x(x)(s(x))}$ is the Bott-Dirac operator of $E_n$ at $t_x(x)(s(x))$.\n\\par\nIt follows from (\\cite{HKT} Proposition 2.13 and Appendix) and (\\cite{Yu00} Proposition 7.7) that the composition $\\alpha\\circ\\beta$ is asymptotically equivalent to\n$\\gamma$ (see also \\cite{SpWi} for a remark on compositions of asymptotic morphisms). Hence, $\\alpha_*\\circ\\beta_*=\\gamma_*$.\n\\par\n{\\it Step 2.} Let $\\delta$ be the asymptotic morphism\n$$\\delta:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk$$\ndefined by the formula\n$$\\delta_t(g\\widehat\\otimes T)= \\Big[ \\, \\Big(    \\big(  \\delta_t(g\\widehat\\otimes T)   \\big)^{(0)}, \\cdots, \\big(  \\delta_t(g\\widehat\\otimes T)   \\big)^{(n)}, \\cdots     \\Big) \\Big]$$\nfor all $g\\in \\cS$ and $\\ttt\\in \\cupdxn$, where\n$$\\Big(  \\delta_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y) = g_{t^2}\\big(  B_{E_n, 0}  \\big)\\widehat\\otimes T^{(n)}(x, y)$$\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$, and $B_{E_n, 0}$ is the Bott-Dirac operator of $E_n$ at the origin $0\\in E_n$.\n\\par\nFor each $x\\in \\zdn, \\nn$, let $U_x:\\, L^2_n\\to L^2_n$ be the unitary operator defined by\n$$\\Big( U_x\\xi \\Big)(v)=\\xi\\Big(  v-t_x(x)(s(x)) \\Big)$$\nfor all $\\xi\\in L^2_n:=L^2(E_n, \\Cliff(E_n))$ and $v\\in E_n$. Then\n$$U_x^{-1} \\, B_{E_n, t_x(x)(s(x))} \\, U_x \\, = \\,  B_{E_n, 0} \\,\\,.$$\n\\par\nFor each $s\\in [0, 1]$, define an asymptotic morphism\n$$\\Phi^{(s)}:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk  \\widehat\\otimes \\cM_2(\\mathbb{C})          $$\nby the formula\n\\[\n\\Big(  \\Phi^{(s)}_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y)   =  \\Big( U_x^{(s)}\\Big)^{-1}\n\t\t\\left[\n\t\t\\begin{array}{cc}\n\t\t\t\\Big(  \\gamma_t(g\\widehat\\otimes T)   \\Big)^{(n)}\\big(x, y\\big)     &      0                 \\\\\n\t\t\t\t0       &       0\n\t\t\\end{array}\n\t\t\\right]\nU^{(s)}_x\n\\]\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$, where\n\\[\nU^{(s)}_x =R(s) \\left[\n\t\t\t\t\t\\begin{array}{cc}\n\t\t\t\t\t\tU_x\\widehat\\otimes 1     &       0          \\\\\n\t\t\t\t\t\t\t\t\t0        &       1\n\t\t\t\t\t\\end{array}\n\t\t\t\\right]\n\t\t R(s)^{-1},  \\quad\nR(s)= \\left[\n\t\t\t\t\t\\begin{array}{cc}\n\t\t\t\t\t\t\\cos(\\frac{\\pi}{2}s)        &       \\sin(\\frac{\\pi}{2}s)          \\\\\n\t\t\t\t\t     -\\sin(\\frac{\\pi}{2}s)        &       \\cos(\\frac{\\pi}{2}s)\n\t\t\t\t\t\\end{array}\n\t\t\t\\right].\n\\]\nThen $\\Phi^{(s)}$ is a homotopy between the asymptotic morphisms $\\gamma=\\Phi^{(1)}$ and $\\delta=\\Phi^{(0)}$. Hence, $\\gamma_*=\\delta_*$.\n\\par\n{\\it Step 3.} For each $\\nn$, let $p^{(n)}$ be the projection of $E_n$ onto the 1-dimensional kernel of the Bott-Dirac operator $B_{E_n, 0}$ at the origin $0\\in E_n$.\nFor each  $s\\in [0, 1]$, define an asymptotic morphism\n$$\\Psi^{(s)}:\\, \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk   $$\nby the formula\n\\[\n\\Big(  \\Psi^{(s)}_t(g\\widehat\\otimes T)   \\Big)^{(n)}(x, y) =\\left\\{\n\\begin{array}{ll}\ng_{t^2}\\big( \\frac{1}{s} B_{E_n, 0}  \\big) \\widehat\\otimes T^{(n)}(x, y),    &     \\mbox{ if } s\\in (0, 1];       \\\\\ng(0)\\cdot p^{(n)} \\widehat\\otimes T^{(n)}(x, y), &   \\mbox{ if } s=0,\n\\end{array}\n\\right.\n\\]\nfor all $t\\geq 1$, $x, y\\in \\zdn, \\nn$.\nThen $\\Psi^{(s)}$ is a homotopy between the asymptotic morphism $\\delta$ and the $*$-homomorphism\n$$\\sigma:\\,  \\cS\\widehat\\otimes \\cumpdxn \\rightarrow \\cumpdxnk $$\ndefined by the formula\n$$\\sigma \\Big( g\\widehat\\otimes \\big[\\big(  T^{(0)}, \\cdots, T^{(n)}, \\cdots    \\big)\\big] \\Big)\n\t=g(0)\\cdot  \\Big[\\Big(  p^{(0)}\\widehat\\otimes T^{(0)}, \\cdots,  p^{(n)}\\widehat\\otimes T^{(n)}, \\cdots    \\Big)\\Big] .$$\nIt is clear that $\\sigma$ induces identity on $K$-theory. Therefore, we conclude that\n$$\\alpha_*\\circ\\beta_*=\\gamma_*=\\delta_*=\\sigma_*=identity$$\non $K_*\\Big(  \\cumpdxn    \\Big)$.\nThe case for $(\\alpha_L)_*\\circ(\\beta_L)_*$ is similar.\n\\hfill{$\\Box$}\n\\par\n\\vskip 5mm\n\\par\n{\\bf Summary of Proof of Theorem 1.1.}  It follows from Theorem 7.11 and  Theorem 6.1, together with an argument of diagram chasing, that\nthe evaluation map\n\\begin{small}\n$$e_*: \\lim_{d\\to \\infty} K_*\\Big( \\culmpdxn \\Big) \\longrightarrow \\lim_{d\\to \\infty} K_*\\Big( \\cumpdxn \\Big) $$\n\\end{small}\nis an isomorphism for a sequence of finite metric spaces $(X_n)_\\nn$ with uniform bounded geometry such that the coarse disjoint union $X=\\bigsqcup_\\nn X_n$ admits a fibred coarse embedding\ninto Hilbert space. That is, Theorem 4.4 holds, which implies Proposition 4.6 and, subsequently, Theorem 1.1.\n\\hfill{$\\Box$}\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nM dwarfs are popular targets for exoplanet research. First, radial velocity (RV) variations induced by the planets around M dwarfs are more significant than those around solar-like stars, making it possible to obtain precise mass measurement towards the terrestrial planet end of the mass distribution. Second, their small stellar radii lead to a large planet-to-star radius ratio, which favors transit detections and further photometric follow-up observations. Planets around M dwarfs are also attractive sources for atmospheric characterization through transmission or emission spectroscopy \\citep{Kempton2018,Batalha2018} as they yield a higher signal-to-noise ratio than equivalent systems with other types of hosts (e.g., LHS 3844b, \\citealt{Vanderspek2019,Kreidberg2019,Diamond2020}). Finally, due to the low stellar luminosity (typically $L<0.1\\ L_{\\odot}$), the habitable zone of M dwarfs is closer to the host star when compared with luminous stars (e.g., TOI-700d, \\citealt{Gilbert2020,Rodriguez2020}), which offers particular advantages to look for planets with potential biosignatures. \n\nOver the last two decades, more than a thousand transiting giant planets (defined as $M_{p}>0.3\\ M_{J}$) have been discovered thanks to successful ground-based surveys, including HATNet \\citep{Bakos2004}, SuperWASP \\citep{Pollacco2006}, KELT \\citep{Pepper2007,Pepper2012} and NGTS \\citep{Chazelas2012,Wheatley2018} as well as space transit missions like {\\it CoRoT} \\citep{Baglin2006}, {\\it Kepler} \\citep{Borucki2010} and {\\it K2} \\citep{Howell2014}. However, even though M dwarfs are the most abundant stellar population in our Milky Way \\citep{Henry2006}, only five giant planets have been confirmed to transit them: Kepler-45b \\citep{Johnson2012}, HATS-6b \\citep{Hartman2015}, NGTS-1b \\citep{Bayliss2018}, HATS-71b \\citep{Bakos2020} and TOI-1899b \\citep{Canas2020}. The deficiency of such systems is thought to be caused by the failed growth of a massive core to start runaway accretion before the gaseous protoplanetary disk dissipates due to the low surface density \\citep{Laughlin2004,Ida2005,Kennedy2008,Liu2020}. Indeed, previous statistical studies of the occurrence rates from {\\it Kepler}\\ show that planets with radii between $1R_{\\oplus}$ and $4R_{\\oplus}$ are frequent around low-mass stars \\citep{Dressing2013,Dressing2015,Hardegree-Ullman2019}. Some of these small planets are possibly the bare cores of failed gas giants. Nevertheless, microlensing surveys have found plenty of cold Jupiters ($a\\gtrsim 1$ AU) around M dwarfs (e.g., \\citealt{Zang2018}), which hints that outer giant planets are probably not rare. A handful of such cases have also been reported by long-term RV observations (e.g., GJ 876b, \\citealt{Marcy2001}; GJ 849b, \\citealt{Butler2006}; GJ 179b, \\citealt{Howard2010}; HIP 79431b, \\citealt{Apps2010}). Gravitational instability is speculated to be the alternative formation mechanism responsible for the surprising number of long-period gas giants around M dwarfs \\citep{Boss2000,Morales2019}. But it is still unclear how these short-period ($P\\lesssim 30$ d) transiting gas giants were formed, and whether they have migrated into their current orbits due to the lack of such systems. Therefore, establishing a well-characterized sample of this kind of planet is an important step to study their formation. Further Rossiter-McLaughlin \\citep{Rossiter1924,McLaughlin1924} or Doppler tomography \\citep{Marsh2001} measurements could reveal the obliquity of these systems, providing important clues about the dynamical history of the planets (e.g., \\citealt{Albrecht2012}).\n\nThe Transiting Exoplanet Survey Satellite ({\\it TESS}, \\citealt{Ricker2014,Ricker2015}), which has performed a two-year all-sky survey, offers exciting opportunities to increase the number of transiting giant planets around M dwarfs. Although {\\it TESS}\\ has already identified several such planet candidates, the intrinsic faintness of their hosts ($V\\gtrsim15$ mag) challenges most ground-based optical spectroscopic facilities to further conduct detailed RV follow-up observations. Some efforts have already been made to validate those planets through multi-color transit modeling and phase curve analysis (e.g., TOI-519b, \\citealt{Parviainen2021}). The new-generation near-infrared spectrograph SPIRou on the Canada-France-Hawaii-Telescope (CFHT) opens a new window to characterize planets around faint stars \\citep{Artigau2014,Donati2020}. It was designed to perform high-precision velocimetry and spectropolarimetry studies. Early observations from SPIRou have shown that it can reach $2\\sim10$ m\/s precision for stars with $H<10$ mag \\citep{Moutou2020,Klein2021,Artigau2021}. Although simulations predict that SPIRou could reach $<2$ m\/s RV precision for inactive M dwarfs with $J<10$ mag (see Figure 5 in \\citealt{Cloutier2018}), precision for faint stars has yet to be determined observationally.\n\nHere we report the discovery of a new transiting giant planet around an M-dwarf star {TOI-530}. We present RV measurements from SPIRou that allow us to obtain a precise companion mass and thus confirm its planetary nature. The rest of the paper is organized as follows. We describe all space and ground-based observational data used in this work in Section \\ref{observations}. Section \\ref{stellar_properties} presents the stellar properties. In Section \\ref{analysis}, we show our analysis of the light curves and RV data. We discuss the prospects of future atmospheric characterization of {TOI-530} b and its potential formation channel in Section \\ref{discussion}. We conclude with our findings in Section \\ref{conclusion}.\n\n\n\n\n\n\n\n\\section{Observations}\\label{observations}\n\\subsection{{\\it TESS}\\ photometry}\n{TOI-530}\\ was observed by {\\it TESS}\\ on its Camera 1 with the two-minute cadence mode in Sector 6 during the primary mission and Sector 33 during the extended mission. The current data span from 2018 December 15th to 2021 January 13th, consisting of 14830 and 17547 measurements, respectively. The target will be revisited in Sectors 44-45 between 2021 October 12th and 2021 December 2nd. Figure \\ref{fov} shows the POSS2 and {\\it TESS}\\ images centered on {TOI-530}.\n\nThe photometric data from Sector 6 were initially reduced by the Science Processing Operations Center (SPOC; \\citealt{Jenkins2016}) pipeline, developed based on the {\\it Kepler}\\ mission's science pipeline. The simple aperture photometry (SAP) flux time series was corrected for instrumental and systematic effects, and for crowding and dilution with the Presearch Data Conditioning (PDC; \\citealt{Stumpe2012,Smith2012,Stumpe2014}) module. Transit signals were searched using the Transiting Planet Search \\citep[TPS;][]{Jenkins2002,Jenkins2017} algorithm on 17 February 2019, yielding a strong transit signal at a period of $\\sim$6.39 days and a transit duration of $\\sim$2.5 hours. The transit signature and pixel data passed all the  validation tests \\citep{Twicken2018,Li2019,Guerrero2021}, including locating the source of the transit signature to within 1 - 3 arcsec of the target star, and no further transiting planet signatures were identified in a search of the residual light curve. The vetting results were reviewed by the TESS Science Office (TSO) and issued an alert for {TOI-530} b as a planet candidate on 28 March 2019.\n\nWe downloaded the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) light curve from the Mikulski Archive for Space Telescopes\\ (MAST\\footnote{\\url{http:\/\/archive.stsci.edu\/tess\/}}) using the \\code{lightkurve} package \\citep{lightkurvecollaboration,lightkurve}. Combining the datasets of two sectors, we conducted an independent transit search by utilizing the Transit Least Squares (TLS; \\citealt{Hippke2019}) algorithm, which is an advanced version of Box Least Square (BLS; \\citealt{Kovacs2002}), after smoothing the full light curve with a median filter. We recovered the 6.387 d transits with a signal detection efficiency (SDE) of $\\sim50$. After subtracting the TLS model from the {\\it TESS}\\ data, we did not find any other significant transit signals existing in the light curve. We detrended the raw {\\it TESS}\\ light curve by fitting a Gaussian Process (GP) model with a Mat\\'{e}rn-3\/2 kernel using the \\code{celerite} package \\citep{Foreman2017}, after masking out all in-transit data. We show the reprocessed light curve in Figure \\ref{transit_detrend}. \n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.94\\columnwidth]{fov.pdf}\n\\includegraphics[width=1.06\\columnwidth]{TPF_Gaia_TIC387690507_S33.pdf}\n\\caption{{\\it Left panel}: The POSS2 blue image of {TOI-530}\\ taken in 1996. The center red dot is the target star in this image and the cyan circle shows its current position, which rules out the unassociated distant eclipsing binary scenario. {\\it Right panel}: Target pixel file (TPF) of {TOI-530}\\ in {\\it TESS}\\ Sector 33 (created with \\code{tpfplotter}, \\citealt{Aller2020}). Different sizes of red circles represent different magnitudes in contrast with {TOI-530}\\ ($\\Delta m$). The aperture used to extract the photometry is overplotted with a red-square region.} \n\\label{fov}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{TOI530_total_lc.png}\n\\caption{{\\it Top panels}: The original {\\it TESS}\\ SAP light curves of {TOI-530}\\ from Sector 6 and 33. {\\it Middle panels}: The PDCSAP light curves of {TOI-530}\\ along with the best-fit GP model shown as red solid lines. {\\it Bottom panels}: The detrended PDCSAP light curves. The transits of {TOI-530} b are marked in blue ticks.} \n\\label{transit_detrend}\n\\end{figure*}\n\n\\subsection{Ground-Based photometry}\\label{gbp}\nWe collected a series of ground-based observations of {TOI-530}, as part of the TESS Follow-up Observing Program (TFOP\\footnote{\\url{https:\/\/tess.mit.edu\/followup}}), to (1) confirm the transit signal on target and rule out nearby eclipsing binary scenario; (2) examine the chromaticity; and (3) refine the transit ephemeris and radius measurement. These observations were scheduled with the help of the {\\it TESS}\\ Transit Finder (\\code{TTF}), which is a customized version of the \\code{Tapir} software package \\citep{Jensen2013}. Due to the observational constraints, unfortunately, we only covered the egress of the event. We summarize the details in Table \\ref{po} and describe individual observations below. We show the raw and detrended ground-based light curves in Figure \\ref{ground_transit_detrend} (see Section \\ref{joint_fit_TESS_ground}).\n\n\\subsubsection{El Sauce}\\label{el}\nAn egress was observed on UT 2019 November 21 in the $R_{c}$ band using the Evans telescope (0.36 m) at the El Sauce Observatory, Chile. The STT 1603 camera has a pixel scale of $\\rm 1.47''$ per pixel. We acquired a total of 65 images over 205 minutes. Photometric analysis was carried out using \\code{AstroImageJ} \\citep{Collins2017} with an uncontaminated aperture of $5.88''$. We excluded all nearby stars within $1'$ as the source causing the {\\it TESS}\\ signal with brightness difference down to $ \\Delta T \\sim 4.1$ mag, and confirmed the signal on target.\n\n\\subsubsection{MuSCAT2}\\label{muscat2}\nWe observed an egress of TOI-530b on the night of UT 2020 January 4 with the multicolor imager MuSCAT2 \\citep{2019JATIS...5a5001N} mounted on the 1.52 m Telescopio Carlos S\\'{a}nchez at Teide Observatory, Tenerife, Spain. MuSCAT2 has a field of view of $7.4' \\times 7.4'$ with a pixel scale of $0.44''$ pixel$^{-1}$ and is able to obtain simultaneous photometry in four bands ($g$, $r$, $i$, and $z_s$). The observations were made with the telescope in optimal focus and the exposure times for each band were 45 s for $g$, 30 s for $r$ and $i$, and 20 s for $z_s$ band. The data were calibrated using standard procedures (dark and flat calibration). Aperture photometry and transit light curve fit was performed using MuSCAT2 pipeline (\\citealp{2020A&A...633A..28P}); the pipeline finds the aperture that minimizes the photometric dispersion while fitting a transit model including instrumental systematic effects present in the time series.  \n\n\\subsubsection{MuSCAT}\\label{muscat}\nWe observed an egress of TOI-530b on UT 2020 March 2 in $g$, $r$, and $z_s$ bands, using the multiband imager MuSCAT \\citep{2015JATIS...1d5001N} mounted on the 188~cm telescope of National Astronomical Observatory of Japan at the Okayama Astro-Complex, Japan. MuSCAT has three CCD cameras, each having a pixel scale of $0.361''$ pixel$^{-1}$ and a field of view of $6.1' \\times 6.1$. We acquired 321, 268, and 474 images with exposure times of 30, 30, and 20~s in $g$, $r$, and $z_s$ bands, respectively. The data were dark-subtracted and flat-field corrected in a standard manner. Aperture photometry was then performed on the reduced images using a custom pipeline \\citep{2011PASJ...63..287F}. The radius of the photometric aperture was chosen to be 18 pixels ($6.5''$) for all bands so that the photometric dispersion was minimized.\n\n\\begin{table*}\n    \\centering\n    \\caption{Ground-based photometric follow-up observations for {TOI-530}}\n    \\begin{tabular}{ccccccccc}\n        \\hline\\hline\n        Telescope &Camera &Filter &Pixel Scale & Aperture Size (pixel) &Coverage      &Date &Duration (minutes) &Total exposures  \\\\\\hline\n        El Sauce (0.36 m) &STT 1603 &$R_{c}$ &1.47 &4 &Egress &2019 November 21 &183 &59\\\\\\hline\n        TCS (1.52 m) &MuSCAT2 &$g$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &305\\\\\n        TCS (1.52 m) &MuSCAT2 &$r$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &456\\\\\n        TCS (1.52 m) &MuSCAT2 &$i$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &456\\\\\n        TCS (1.52 m) &MuSCAT2 &$z_{s}$ &0.44 &9.8 &Egress &\t2020 January 4 &237 &238\\\\\\hline\n        NAOJ (1.88 m) &MuSCAT &$g$ &0.36 &18 &Egress &2020 March 2 &177 &321\\\\\n        NAOJ (1.88 m) &MuSCAT &$r$ &0.36 &18 &Egress &2020 March 2 &177 &268\\\\\n        NAOJ (1.88 m) &MuSCAT &$z_{s}$ &0.36 &18 &Egress &2020 March 2 &177 &474\\\\\n         \\hline\n    \\end{tabular}\n    \\label{po}\n\\end{table*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{ground_only.pdf}\n\\caption{Ground-based light curves for all available instrument. The blue dots are the raw data while the black solid line represents the best fit GP+transit model. The black dots are results after subtracting the GP model (i.e., detrended data). We use these detrended light curves in the final joint-fit (see Section \\ref{joint}).} \n\\label{ground_transit_detrend}\n\\end{figure*}\n\n\\subsection{Spectroscopic Observations}\n\\subsubsection{IRTF}\nWe observed TOI-530 on UT 2019 April 23 with the uSpeX spectrograph \\citep{2003PASP..115..362R, 2004SPIE.5492.1498R} on the 3-m NASA Infrared Telescope Facility (IRTF). Our data was collected in the SXD mode using the $0\\farcs3 \\times 15\\arcsec$ slit and covers a wavelength range of $0.7-2.55$ $\\mu$m. The data was reduced using the \\texttt{Spextool} pipeline \\citep{2004PASP..116..362C}. After reducing, we RV-correct our spectrum using \\texttt{tellrv} \\citep{2014AJ....147...20N}, with which we estimate a systemic radial velocity of $-26 \\pm 5$ km\/s. By comparing our spectrum to those provided by the IRTF library \\citep{Rayner2009}, we determine that our spectrum best matches that of a star of spectral type M0.5V. Lastly, we calculate the metallicity of TOI-530 following the relations defined in \\cite{Mann2013} for cool dwarfs with spectral types between K7 and M5. In performing this calculation, we opted to only use the $Ks$-band spectrum, as \\cite{2019AJ....158...87D} found $Ks$-band spectra to produce more reliable metallicities and suffer less telluric contamination than $H$-band spectra. Our analysis yield metallicities of [Fe\/H] = $0.376 \\pm 0.095$ and [M\/H] = $0.218 \\pm 0.092$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{SpeX_spectrum.pdf}\n\\caption{Renormalized SpeX spectrum of {TOI-530}\\ (red line) and the comparison spectrum (blue line) taken from the IRTF library \\citep{Rayner2009}. The strong atomic features are marked based on the results from \\citet{Cushing2005}. The NIR spectrum of {TOI-530}\\ corresponds to a spectral type of M0.5V.} \n\\label{IRTF}\n\\end{figure}\n\n\\subsubsection{CFHT\/SPIRou}\nWe monitored {TOI-530}\\ over 5 epochs between UT 2020 September 26 and UT 2020 October 5 using SPIRou (standing for SpectroPolarim\\`etre InfraROUge), which is a new-generation high resolution ($64,000$) fiber-fed spectrograph with polarimetric and precision velocimetry capacities, installed at CFHT in 2018 \\citep{Artigau2014,Donati2018}. It has a large bandwidth (from 0.95 to 2.5 $\\mu$m) allowing the detection of several stellar lines in a single shot thus enhancing the precision of the measurement of the stellar radial velocity. For each night, we obtained three sequences, with 975s exposure time for each. The spectroscopic data were reduced using the standard data reduction pipeline (APERO, Cook et al. in prep), which performs the data calibration and corrects the telluric and night-sky emission \\citep{Artigau2021}. For the Night-sky emission, it is corrected using a principal component analysis (PCA) model of OH emission constructed from a library of high-SNR sky observations \\citep{Artigau2014PCA}. The telluric absorption is corrected using a PCA-based approach on residuals after fitting for a basic atmospheric transmission model (TAPAS, \\citealt{Bertaux2014}).\n\nWe extracted the RVs of TOI-530 from the telluric-subtracted SPIRou spectra using \\code{wobble} \\citep{Bedell2019}. Briefly, \\code{wobble} constructs a linear model to infer the stellar and time-varying telluric spectra without requiring any prior knowledge on them, while solving for the RV at each epoch. \n\nWe only used the orders 29--37 (around 1490--1800~nm, all in the H band) to extract RVs for the following reasons. We dropped the orders 13--17, 23--28, 39--43, and 47--49 (around 1130--1230~nm, 1330--1490~nm, 1850--2010~nm, 2326--2510~nm, respectively), because the telluric absorption lines are too heavy. These orders with heavy telluric absorption are basically the wavelength regions in between the photometric bands (Y, J, H, K, and at the end of the K band). Furthermore, we dropped the orders 1--12, 18--22, 44--46 (around 965--1140~nm, 1225--1340~nm, and 2132--2290~nm, respectively), because their signal-to-noise ratios (SNRs) are too low (lower than $\\sim$30). The low SNRs caused poor corrections of the telluric lines by the SPIRou pipeline, which manifests as significant residuals of telluric emission\/absorption, as well as some abnormal features caused by improper telluric subtraction. In addition, the authors of \\code{wobble} also cautioned regarding applying the code to spectral data with SNR less than 50 \\citep[][e.g., in their example using the Barnard's star's data]{Bedell2019}. We extracted RVs from the low-SNR orders and saw little RV variations in these orders, which we believe is due to the fact that \\code{wobble} struggles to recover any RV information from these low-SNR spectra with heavy telluric residuals. \n\nTo pre-process the spectra, we first dropped 500 pixels at both edges of each order and masked out the occasional residual emission lines not fully subtracted by the SPIRou pipeline as follows: We calculated the 80th percentile of the flux (denoted as $q$) in any given order, labeled all pixels with flux larger than $3 \\times q$ as the emission-line pixels, and masked out 5 pixels in total centered around them. Then we scaled the blaze function offered by the SPIRou pipeline to the flux level of the observed spectrum and calculated the flux minus the blaze function at each pixel. If the difference is more than half of the maximum of the blaze function of corresponding order, such pixels were considered as emission-line pixels, and 5 pixels around them were masked out. Then, we used the scaled blaze function to continuum-normalize the spectra. \n\nNext, we passed the natural log of the wavelength, the natural log of the flux, the estimated inverse variance of the flux (set as photon counts at each pixel, assuming Poisson noise on the flux), the time of the observations, the BERVs and the airmass values to \\code{wobble}. We let \\code{wobble} only infer the stellar the spectra to extract RVs because the SPIRou pipeline already divided out telluric absorption. When \\code{wobble} infers the stellar spectrum, it needs optimized L1 and L2 regularization parameters for each orders. For simplicity, we set these regularization parameters to the default values in the \\code{wobble} code, which are the same for all orders.\n\nTo validate our work, we divided each order into the left part and the right part so that the total photon counts of each part are equal. Then we used the same method to get RVs from each part respectively. Comparing the RVs from the left and the right parts of each order, we found that the differences are on par with the RV differences between the three observations taken on the same night (i.e., the intra-night RV variation as derived using the full order). The RV signals are basically consistent cross the nine orders we analyzed. This suggests that our results are unlikely to arise from random noise but instead are of real astrophysical origin. However, we found that the differences between the RVs reported from the left or the right parts of each order (typically 10--30 m\/s) are significantly larger than the RV error bars reported by \\code{wobble} (typically 1.6--1.9 m\/s). Therefore, we calculated the standard deviation of the six RVs from the left and the right of each night (two RVs per observation, 3 observations per night) and used them as the more realistic estimates of the uncertainties of the RVs, which are what we present in Table~\\ref{spirourv}.\n\n\\begin{comment}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{SPIRou_RV_three_part.png}\n\\caption{RVs from the left part and the right part compared with the RVs from the whole spectra. To be clear, the BJDs of the left part and the right part are slightly changed. \\todo{Tianjun: This is an important result. Let's make it a two-column plot. The x axis needs some updates (BJD-2457000). Q: there are no RV error bars for each data point? Mention that error bars are too small and unrealistic so did not plot.}} \n\\label{SPIRou_validation}\n\\end{figure*}\n\\end{comment}\n\n\n\\begin{table}\n     \\centering\n     \\caption{SPIRou RV measurements of {TOI-530}. Each observation took an exposure time of 975s. The RV offset here is arbitrary.}\n     \\begin{tabular}{ccc}\n         \\hline\\hline\n         BJD$_\\mathrm{TDB}$       &RV\\ (m~s$^{-1}$) &$\\sigma_{\\rm RV}$\\ (m~s$^{-1}$) \\\\\\hline\n         2459119.08206 \t&29356.43 \t&20.14 \\\\ \n         2459119.09361 \t&29372.09 \t&20.14 \\\\ \n         2459119.10515 \t&29389.86 \t&20.14 \\\\\n         2459120.06581 \t&29486.04 \t&12.55 \\\\\n         2459120.07736 \t&29497.80 \t&12.55 \\\\\n         2459120.08897 \t&29500.96 \t&12.55 \\\\\n         2459123.06563 \t&29342.45 \t&14.13 \\\\\n         2459123.07718 \t&29358.37 \t&14.13 \\\\\n         2459123.08873 \t&29364.25 \t&14.13 \\\\\n         2459127.07895 \t&29461.84 \t&20.86 \\\\\n         2459127.09089 \t&29495.09 \t&20.86 \\\\\n         2459127.10289 \t&29503.15 \t&20.86 \\\\\n         2459128.06393 \t&29353.01 \t&31.13 \\\\\n         2459128.07548 \t&29374.36 \t&31.13 \\\\\n         2459128.08703 \t&29445.57 \t&31.13 \\\\\n          \\hline\n     \\end{tabular}\n    \n    \n    \n     \\label{spirourv}\n\\end{table}\n\n\\subsection{High Angular Resolution Imaging}\nIf an exoplanet host star has a spatially close companion, that companion (bound or line of sight) can create a false-positive transit signal if it is, for example, an eclipsing binary (EB). For small stars and large planets, this is an especially important check to make, due to the paucity of giant planets orbiting M stars. ``Third-light\" flux from the close companion star can lead to an underestimated planetary radius if not accounted for in the transit model \\citep{Ciardi2015} and cause non-detections of small planets residing with the same exoplanetary system \\citep{Lester2021}. Additionally, the discovery of close, bound companion stars, which exist in nearly one-half of FGK type stars \\citep{Matson2018} and less so for M class stars, provides crucial information toward our understanding of exoplanetary formation, dynamics and evolution \\citep{Howell2021}. Thus, to search for close-in bound companions unresolved in TESS or other ground-based follow-up observations, we obtained high-resolution imaging observations of TOI-530.\n\n\\subsubsection{Keck\/NIRC2 Adaptive Optics Imaging}\nWe observed {TOI-530}\\ with infrared high-resolution adaptive optics (AO) imaging at Keck Observatory \\citep{Ciardi2015,Schlieder2021} on UT 2019 April 7. The observations were made with the NIRC2 instrument on Keck-II behind the natural guide star AO system. The standard 3-point dither pattern was used to avoid the left lower quadrant of the detector which is typically noisier than the other three quadrants. The dither pattern step size\nwas 3$''$ and it was repeated twice, with each dither offset from the previous one by 0.5$''$.\n\nThe observations were taken in the broad-band K ($\\lambda_{o}=2.1956$ $\\mu$m; $\\Delta \\lambda=0.336$ $\\mu$m) with an integration time of 4 s per frame for a total on-source integration time of 36 s. The camera was in the narrow-angle mode with a full field of view of 10$''$ and a pixel scale of approximately 0.009942$''$ per pixel. The Keck AO observations show no additional stellar companions were detected to within a resolution $\\sim0.056''$ FWHM. The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around the primary target every $20^{\\circ}$ at radial separations of integer multiples of the FWHM of the central source \\citep{Furlan2017}. The brightness of each injected source was scaled until standard aperture photometry detected it with $5\\sigma$ significance. The resulting brightness of the injected sources relative to the target {TOI-530}\\ was regarded as the contrast limits at that injection location. The final $5\\sigma$ limit at each separation was determined from the average of all of the determined limits at that separation while the uncertainty was given by the RMS dispersion of the results for different azimuthal slices at a given radial distance. We show the 2$\\mu$m sensitivity curve in the left panel of Figure \\ref{imaging} along with an inset image zoomed to primary target, which shows no other companion stars. \n\n\\subsubsection{Gemini-North Speckle Imaging}\nTOI-530 was observed on 2020 February 17 UT using the 'Alopeke speckle instrument on the Gemini North 8-m telescope\\footnote {\\url{https:\/\/www.gemini.edu\/sciops\/instruments\/alopeke-zorro\/}}.  'Alopeke provides simultaneous speckle imaging in two bands (562nm and 832 nm) with output data products including a reconstructed image with robust contrast limits on companion detections (e.g., \\citealt{Howell2016}). Ten sets of $1000\\times0.06$ sec exposures were collected and subjected to Fourier analysis in our standard reduction pipeline \\citep{Howell2011}. The right panel of Figure \\ref{imaging} shows our final contrast curves and the 832 nm reconstructed speckle image. We find that TOI-530 is a single star with no companion brighter than 5-6 magnitudes below that of the target star (earlier than $\\sim$ M4.5V) from the diffraction limit (20 mas) out to 1.2$''$. At the distance of TOI-530 (d=148 pc) these angular limits correspond to spatial limits of 3 to 178 au.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{high_resolution_plot.pdf}\n\\caption{{\\it Left panel}: NIRC2 AO image (inset) and $\\rm K_{s}$-band contrast curve for {TOI-530}. The black line is the $5\\sigma$ sensitivity limit. The shaded purple region represents the azimuthal dispersion ($1\\sigma$) of the contrast determinations. {\\it Right panel}: The $5\\sigma$ 'Alopeke speckle imaging contrast curves in both filters as a function of the angular separation out to 1.2 arcsec, the end of speckle coherence. The inset shows the reconstructed 832 nm image with a 1 arcsec scale bar. The star, TOI-530, was found to have no close companions to within the contrast levels achieved. } \n\\label{imaging}\n\\end{figure*}\n\n\n\\section{Stellar Characterization}\\label{stellar_properties}\n\nWe first use 2MASS observed $m_{K}$ and the parallax from {\\it Gaia}\\ EDR3 to calculate the absolute magnitude, of which we obtain $M_{K}= 5.42 \\pm 0.13$ mag. We then estimate the stellar radius following the polynomial relation between $R_{\\ast}$ and $M_{K}$ derived by \\cite{Mann2015}, and we find $R_{\\ast}=0.54\\pm0.02\\ R_{\\odot}$, assuming a typical uncertainty of 3\\% (see Table 1 in \\citealt{Mann2015}). For comparison, we also estimate the stellar radius $R_{\\ast}=0.55\\pm0.03 R_{\\odot}$ based on the angular diameter relation in \\cite{Boyajian2014}, consistent with our previous estimate within $1\\sigma$. \n\nUsing the empirical polynomial relation between bolometric correction ${\\rm BC}_{K}$ and $V-J$ in \\cite{Mann2015}, we find ${\\rm BC}_{K}$ to be $2.60\\pm0.13$ mag. Thus, we derive a bolometric magnitude $M_{\\rm bol}=8.02\\pm 0.13$ mag, leading to a bolometric luminosity of $L_{\\ast}=0.049\\pm0.005\\ L_{\\odot}$. To estimate the stellar effective temperature of {TOI-530}, we first take use of the Stefan-Boltzmann law. Coupled with the aforementioned stellar radius and bolometric luminosity we derived, we get $T_{\\rm eff}=3666\\pm146$ K. As an independent check, we then obtain $T_{\\rm eff}$ following the empirical relation reported by \\cite{Mann2015} and we find $T_{\\rm eff}=3650\\pm100$ K. Both estimations agree well with the result $T_{\\rm eff}=3663\\pm124$ K from \\cite{Pecaut2013}. \n\nFinally, we evaluate that {TOI-530}\\ has a mass of $M_{\\ast}=0.53\\pm0.01\\ M_{\\odot}$ using Equation 2 in \\cite{Mann2019} according to the $M_{\\ast}$-$M_{K}$ relation. This is consistent with the value $M_\\ast = 0.52 \\pm 0.03\\ M_\\odot$ given by the eclipsing-binary based empirical relation of \\citet{Torres2010}.\n\nAs an independent check, we carry out an analysis of the broadband Spectral Energy Distribution (SED) together with the {\\it Gaia\\\/} EDR3 parallax in order to determine an independent, empirical measurement of the stellar radius, following the procedures described in \\citet{Stassun2016}, \\citet{Stassun2017}, and \\citet{Stassun2018}. We pull the $JHK_S$ magnitudes from {\\it 2MASS} \\citep{Cutri2003,skrutskie2006}, the W1--W3 magnitudes from {\\it WISE} \\citep{wright2010},  the $grizy$ magnitudes from Pan-STARRS \\citep{Magnier2013}, and three {\\it Gaia}\\ magnitudes $G, G_{\\rm BP}, G_{\\rm RP}$ \\citep{GaiaEDR3}. Together, the available photometry spans the full stellar SED over the wavelength range 0.4\\,--\\,10~$\\mu$m (see Figure~\\ref{fig:sed}). \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth,trim=10 5 20 15,clip]{TOI530_sed.pdf}\n    \\caption{The best SED fit for {TOI-530}. Red symbols represent the observed photometric measurements, where the horizontal bars represent the effective width of the passband. Blue symbols are the model fluxes from the best-fit NextGen atmosphere model (black). \n\\label{fig:sed}}\n\\end{figure}\n\nWe perform a fit using NextGen stellar atmosphere models, with the $T_{\\rm eff}$, $\\log g$, and [Fe\/H] taken from the spectroscopic analysis. The remaining parameter is the extinction ($A_V$), which we limit to the full line-of-sight extinction from the dust maps of \\citet{schlegel1998}. The resulting fit is shown in Figure.~\\ref{fig:sed} with a reduced $\\chi^2$ of 1.6 and best-fit extinction of $A_V = 0.00^{+0.03}_{-0.00}$. Integrating the model SED gives the bolometric flux at Earth of \\mbox{$F_{\\rm bol} =  7.009 \\pm 0.081 \\times 10^{-11}$ erg~s$^{-1}$~cm$^{-2}$}. Taking the $F_{\\rm bol}$ and $T_{\\rm eff}$ together with the {\\it Gaia\\\/} parallax, with no adjustment for systematic parallax offset \\citep[see, e.g.,][]{Stassun2021}, gives the stellar radius as $R_\\ast = 0.547 \\pm 0.030\\ R_\\odot$. \n\nCombining all the results above, we adopt the weighted-mean values of effective temperature $T_{\\rm eff}$, stellar radius $R_{\\ast}$ and stellar mass $M_{\\ast}$ as listed in Table \\ref{starparam}.\n\nTo identify the Galactic population membership of {TOI-530}, we first calculate the three-dimensional space motion with respect to the LSR based on \\cite{Johnson1987}. We adopt the astrometric values ($\\varpi$, $\\mu_{\\alpha}$, $\\mu_{\\delta}$) from {\\it Gaia}\\ EDR3 and the spectroscopically determined systemic RV from the SpeX spectrum, and we find $U_{\\rm LSR}=48.99\\pm4.59$ km s$^{-1}$, $V_{\\rm LSR}=-20.10\\pm1.92$ km s$^{-1}$, $W_{\\rm LSR}=-6.73\\pm0.56$ km s$^{-1}$. Following the procedure described in \\cite{Bensby2003}, we compute the relative probability $P_{\\rm thick}\/P_{\\rm thin}=0.02$ of {TOI-530}\\ to be in the thick and thin disks by taking use of the recent kinematic values from \\cite{Bensby2014}, indicating that {TOI-530}\\ belongs to the thin-disk population. We further integrate the stellar orbit with the ``MWPotential2014'' Galactic potential using \\code{galpy} \\citep{Bovy2015} following \\cite{Gan2020}, and we estimate that the maximal height $Z_{\\rm max}$ of {TOI-530}\\ above the Galactic plane is about $109$ pc, which agrees with our thin-disk conclusion. \n\nWe finally perform a frequency analysis on the {\\it TESS}\\ PDCSAP photometry after masking the known in-transit data using the generalized Lomb-Scargle periodogram \\citep{Zechmeister2009} to look for stellar activity signals. We find a peak at around $9.4$ d in the {\\it TESS}\\ Sector 6 data, which may be attributed to stellar rotation. However, this periodic signal is not significant in the generalized Lomb-Scargle periodogram of the {\\it TESS}\\ photometry taken in the extended mission. We further analyze the ground-based long-term photometry from the Zwicky Transient Facility (ZTF; \\citealt{Masci2019}). ZTF took a total of 273 exposures for {TOI-530}, which spanned 1036 d. We clip outliers above the $3\\sigma$ level and 242 measurements are left. However, we find that the 9.4 d signal does not show up in the corresponding generalized Lomb-Scargle periodogram, either. Additionally, \\citealt{Newton2018} shows a typical rotational period of $\\sim40$ d for a $0.5\\ M_{\\odot}$ star. We thus conclude that the $9.4$ d signal is probably not associated with stellar rotation. Future {\\it TESS}\\ data to be obtained will allow better identification of the correct rotation period of this target. \n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{TESS_GLS_updated.pdf}\n\\caption{Generalized Lomb-Scargle periodograms of the TESS PDCSAP photometry from two sectors. The theoretical FAP levels of 10, 1, and 0.1 percent are marked as horizontal solid, dashed, and dot\u2013dashed lines. The vertical red lines mark the maximum peaks of the periodograms.} \n\\label{TESS_GLS}\n\\end{figure}\n\n\n\n\\begin{table}\\label{stellarinfor}\n   \n    \\caption{Basic information of {TOI-530}}\n    \\begin{tabular}{lll}\n        \\hline\\hline\n        Parameter       &Value       \\\\\\hline\n        \\it{Main identifiers}                    \\\\\n        \n         TOI                     &$530$         \\\\\n         TIC                     &$387690507$   \\\\\n         {\\it Gaia}\\ ID            &$3353218995355814656$ \\\\\n         \\it{Equatorial Coordinates} \\\\\n         $\\rm R.A.\\ (J2015.5)$                    &06:53:39.08 \\\\\n         $\\rm DEC.\\ (J2015.5)$                    &12:52:53.68    \\\\\n         \\it{Photometric properties}\\\\\n         ${\\it TESS}$\\ (mag)           &$13.5287\\pm0.0076$   &$\\rm TIC\\ V8^{[1]}$     \\\\\n         ${\\it Gaia}$\\ (mag)           &$14.6217\\pm0.0006$   &{\\it Gaia}\\ EDR3$^{[2]}$   \\\\\n         {\\it Gaia}\\ BP\\ (mag)           &$15.814\\pm0.004$   &{\\it Gaia}\\ EDR3   \\\\\n         {\\it Gaia}\\ RP\\ (mag)           &$13.538\\pm0.002$   &{\\it Gaia}\\ EDR3   \\\\\n         $B$\\ (mag)                    &$16.708\\pm0.044$         &APASS \\\\\n         $V$\\ (mag)                    &$15.403\\pm0.136$          &APASS \\\\\n         $J$\\ (mag)                    &$12.112\\pm0.023$   &2MASS \\\\\n         $H$\\ (mag)                    &$11.468\\pm0.030$   &2MASS \\\\\n         $K$\\ (mag)                    &$11.238\\pm0.020$    &2MASS \\\\\n         \\wise1 (mag)                   &$11.124\\pm0.023$   &{\\it WISE} \\\\\n         \\wise2 (mag)                   &$11.087\\pm0.020$   &{\\it WISE} \\\\\n         \\wise3 (mag)                   &$10.907\\pm0.139$   &{\\it WISE} \\\\\n         \\wise4 (mag)                   &$8.735\\pm0.429$   &{\\it WISE} \\\\\n         \\it{Astrometric properties}\\\\\n         $\\varpi$ (mas)              &$6.77\\pm0.02$  &{\\it Gaia}\\ EDR3  \\\\\n         $\\mu_{\\rm \\alpha}\\ ({\\rm mas~yr^{-1}})$     &$13.62\\pm0.03$   &{\\it Gaia}\\ EDR3   \\\\\n         $\\mu_{\\rm \\delta}\\ ({\\rm mas~yr^{-1}})$     &$-62.52\\pm0.02$   &{\\it Gaia}\\ EDR3  \\\\\n         RV\\ (km~s$^{-1}$)                          &$-25.93\\pm2.00$ &This work  \\\\\n         \\it{Derived parameters} \\\\\n         Distance (pc)                &$147.7\\pm0.6$  &This work     \\\\\n         $U_{\\rm LSR}$ (km~s$^{-1}$)       &$48.99\\pm4.59$     &This work\\\\\n         $V_{\\rm LSR}$ (km~s$^{-1}$)       &$-20.10\\pm1.92$     &This work\\\\\n         $W_{\\rm LSR}$ (km~s$^{-1}$)       &$-6.73\\pm0.56$     &This work\\\\\n         $M_{\\ast}\\ (M_{\\odot})$ &$0.53\\pm 0.02$ &This work       \\\\\n         $R_{\\ast}\\ (R_{\\odot})$ &$0.54\\pm 0.03$ &This work       \\\\\n         $\\rho_\\ast\\ ({\\rm g~cm^{-3}})$ &$4.74\\pm 1.11$ &This work \\\\\n         $\\log g_{\\ast}\\ ({\\rm cgs})$       &$4.70\\pm 0.03$  &This work        \\\\\n         $L_{\\ast}\\ (L_{\\odot})$ &$0.049\\pm0.005$  &This work    \\\\\n         $T_{\\rm eff}\\ ({\\rm K})$           &$3659\\pm 120$  &This work       \\\\\n         $\\rm [Fe\/H]$  &$0.376\\pm 0.095$ &This work \\\\\n         $\\rm [M\/H]$  &$0.218\\pm0.092$ &This work\\\\\n        \n         \\hline\\hline \n    \\end{tabular}\n    \\begin{tablenotes}\n    \\item[1]  [1]\\ \\cite{Stassun2017tic,Stassun2019tic} \n    \\item[2]  [2]\\ \\cite{GaiaEDR3}\n    \\end{tablenotes}\n    \\label{starparam}\n\\end{table}\n\n\n\\section{Analysis and results}\\label{analysis}\n\\subsection{Photometric Analysis}\\label{transit}\n\\subsubsection{{\\it TESS}\\ only}\\label{tess_only}\nWe first model the detrended {\\it TESS}\\ only photometry by utilizing the \\code{juliet} package \\citep{juliet}, which employs \\code{batman} to build the transit model \\citep{Kreidberg2015}. Dynamic nested sampling is applied in \\code{juliet} to determine the posterior estimates of system parameters using the publicly available package \\code{dynesty} \\citep{Higson2019,Speagle2019}.\n\nWe set uninformative uniform priors on both the transit epoch ($T_{0}$) and the orbital period ($P_{b}$), centered on the optimized value obtained from the TLS analysis. Following the approach described in \\cite{Espinoza2018}, instead of directly fitting for the radius ratio ($p=R_{p}\/R_{\\ast}$) and the impact parameter ($b=a\/R_{\\ast}\\cos i$), we apply the new parametrizations $r_{1}$ and $r_{2}$ to sample points, for which we impose uniform priors between 0 and 1. This new parametrization allows us to only sample physically meaningful values of a transiting system with $0 < b < 1 + p$, which reduces the computational cost. We adopt a quadratic limb-darkening law for the {\\it TESS}\\ photometry, where we place a uniform prior on both coefficients ($q_{1}$ and $q_{2}$, \\citealt{Kipping2013}). Since photometric-only data weakly constrain the orbital eccentricity, we fix $e$ at zero and include a non-informative log-uniform prior on stellar density. We fit an extra flux jitter term to account for additional systematics. As the {\\it TESS}\\ PDCSAP light curve has already been corrected for the light dilution, we fix the dilution factors $D$ to 1. Table \\ref{tess_only_fit_priors} summarizes the prior settings we adopt as well as the best-fit value of each parameter. We then rerun the photometry-only fit with free $e$ and $w$ to examine potential evidences of eccentricity by comparing the Bayesian model log-evidence ($\\ln Z$) difference between the circular and eccentric orbit models calculated using the \\code{dynesty} package. Generally, we consider a model is strongly favored than another if $\\Delta \\ln Z>5$ \\citep{Trotta2008}. We find that the circular orbit model is slightly preferred with a Bayesian evidence improvement of $\\Delta \\ln Z=\\ln Z_{\\rm Circular}-\\ln Z_{\\rm Keplerian}=2.8$. We thus conclude that there is no evidence of orbital eccentricity in the {\\it TESS}\\ time-series data. We use the posteriors from the circular orbit fit as a prior to detrend all ground-based photometric data (see next Section).\n\n\n\n\n\n\n\\subsubsection{Ground-based photometric data}\\label{joint_fit_TESS_ground}\nSince all of the eight ground light curves only covered partial transits, the way of detrending generally correlates with the final modeling results. Therefore, we decide to independently detrend all ground photometry in a uniform way using Gaussian processes. As there are no obvious quasi-periodic oscillations existing in data from different facilities, we choose the Mat\\'{e}rn-3\/2 kernel, formulated as:\n\\begin{equation}\n    k_{i,j}(\\tau) = \\sigma^{2}\\left(1+\\frac{\\sqrt{3}\\tau}{\\rho}\\right){\\rm exp}\\left(\\frac{\\sqrt{3}\\tau}{\\rho}\\right),\n\\end{equation}\nwhere $\\tau$ is the time-lag, and $\\sigma$ and $\\rho$ are the covariance amplitude and the correlation timescale of the GP, respectively. Taking the posteriors from the previous {\\it TESS}\\ only fit into account, we put a constraint on the priors to optimize the sampling and reduce the computational time cost. We list our priors in Table \\ref{ground_only_fit_priors} and show the raw and detrended ground light curves in Figure \\ref{ground_transit_detrend}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{TOI530_phase_lc.pdf}\n\\caption{{\\it Top panel}: Phase-folded TESS photometry of {TOI-530}. The red solid line represents the median posterior model. {\\it Bottom panel}: The residuals of the TESS data after subtracting the best-fit transit model. } \n\\label{TESS_transit}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{TOI530_phase_followup_lc_residual.pdf}\n\\caption{Left panels show unbinned phase-folded follow-up transit light curves of {TOI-530}. The instrument and observational band information is presented at the top left of each panel. Our best-fit models are shown as red solid lines. The residuals are shown in the right panels.} \n\\label{ground_transit}\n\\end{figure}\n\n\\subsection{RV-only modeling}\\label{rv}\nWe carry out a preliminary RV-only fit using \\code{juliet}, which utilizes the \\code{radvel} package to build the Keplerian model \\citep{Fulton2018}. In order to reduce the potential errors induced by the orbital period and timing, we fix $P_{b}$ and $T_{0,b}$ at the best-fit transit ephemeris derived from the previous {\\it TESS}\\ only fit. Due to the limited number of RV points and our previous insignificant detection of eccentricity (see Section \\ref{tess_only}), we fit a circular orbit model with $e$ fixed at zero. Since our RV observations only have a short time span, we do not take the RV slope and quadratic term ($\\dot{\\gamma}$ and $\\ddot{\\gamma}$) into consideration in the RV modeling, and we simply fix them at 0. Thus the remaining degrees of freedom are the RV semi-amplitude $K_{b}$, the systemic velocity $\\mu$ and the extra jitter term $\\sigma$, which is used to account for the additional white noise. We adopt wide uniform priors on $K_{b}$ and $\\mu$ but a log-uniform prior on $\\sigma$. Our model reveals that the SPIRou RVs have a semi-amplitude of $K=67.2\\pm15.1$ m\/s. Table \\ref{rvonly_priors} provides our prior settings and the median value of the posterior of each parameter along with their $1\\sigma$ confidence interval. \n\nWe then construct a flat RV model to test the robustness of our RV detection above. Compared with the flat model, we find that our circular orbit model has a $\\ln Z$ improvement of $\\Delta \\ln Z=\\ln Z_{\\rm Circular}-\\ln Z_{\\rm Flat}=4.8$, supporting a significant RV detection.\n\n\n\n\\begin{comment}\n\\begin{table}\n    \\centering\n    \\caption{Model comparison of RV only fits with juliet.}\n    \\begin{tabular}{cccc}\n        \\hline\\hline\n        Model   &Type     &$\\ln Z$    &$\\Delta \\ln Z$     \\\\\\hline\n        $\\rm M1$    &Flat            &-87.5 &-12.3\\\\\n       \n       \n        $\\rm M2$    &Circular                     &-82.7  &-7.5\\\\\n        $\\bf {M3}$    &{\\bf Eccentric}          &{\\bf -75.2}   &{\\bf 0}\\\\\n       \n       \n        \\hline\\hline \n    \\end{tabular}\n   \n   \n   \n   \n    \\label{modelcomp}\n\\end{table}\n\\end{comment}\n\n\\subsection{Joint RV and transit analysis}\\label{joint}\n\nIn order to obtain precise transit ephemeris and physical parameters, we finally jointly model the detrended {\\it TESS}\\ photometry and all ground-based re-processed light curves together with the SPIRou RVs. We adopt the identical priors on planetary and {\\it TESS}\\ photometry parameters as in Section \\ref{tess_only}. While for the ground photometric data, we choose the linear law to parameterize the limb-darkening effect and put a Gaussian prior on the theoretical estimate derived from the \\code{LDTK} package with a width of 0.1 \\citep{Husser2013,Parviainen2015}. Similarly, we also fit an extra flux jitter term for each ground instrument to account for additional white noise. As there are less contamination in the ground data, we fix all dilution factors $D$ to 1. For the SPIRou radial velocities, we adopt the same priors as the circular orbit model in Section \\ref{rv}. We find the {TOI-530} b has a mass of $0.4\\pm0.1\\ M_J$ with a radius of $0.83\\pm0.05\\ R_{J}$, which is the typical size of a giant planet without much inflation. We show the phase-folded light curves along with the best-fit models in Figures \\ref{TESS_transit} and \\ref{ground_transit}. Figure \\ref{SPIROU} shows the SPIRou data and the best-fit RV model.  Table \\ref{tranpriors} summarizes the priors we set in the final joint fit as well as the best-fit value of each parameter. We list the final derived physical parameters in Table \\ref{physical_parameter}.\n\nSince there are a total of 5 nearby stars of {TOI-530}\\ with $T{\\rm mag}<15.5$ located within $1'$ and the light from the brightest star among them (Gaia DR2 3353218784898973312, $T{\\rm mag}=11.0$; star 5 in the right panel of Figure \\ref{fov}) is expected to have a significant contribution of the contamination flux in the photometric aperture due to the large {\\it TESS}\\ pixel scale ($21''$\/pixel), we rerun the joint fit to examine whether additional dilution correction is needed. We set a Gaussian prior on the {\\it TESS}\\ dilution factor $D_{\\rm TESS}$, centered at 1 with a $1\\sigma$ width of 0.1, and keep the left prior settings the same as above. We obtain $D_{\\rm TESS}=0.97\\pm0.03$ and a radius ratio of $R_{p}\/R_{\\ast}=0.156\\pm0.001$, consistent with the result without considering light correction.\n\n\\begin{table*}\n   \n    \\caption{Prior settings and the best-fit values along with the 68\\% credibility intervals in the final joint fit for {TOI-530}. $\\mathcal{N}$($\\mu\\ ,\\ \\sigma^{2}$) means a normal prior with mean $\\mu$ and standard deviation $\\sigma$. $\\mathcal{U}$(a\\ , \\ b) stands for a uniform prior ranging from a to b. $\\mathcal{J}$(a\\ , \\ b) stands for a Jeffrey's prior ranging from a to b.}\n    \\begin{tabular}{lccr}\n        \\hline\\hline\n        Parameter       &Prior &Best-fit    &Description\\\\\\hline\n        \\it{Planetary parameters}\\\\\n       \n        $P_{b}$ (days)  &$\\mathcal{U}$ ($6.2$\\ ,\\ $6.6$)  &$6.387597^{+0.000019}_{-0.000018}$\n        &Orbital period of {TOI-530} b.\\\\\n        $T_{0,b}$ (BJD-2457000)  &$\\mathcal{U}$ ($1468$\\ ,\\ $1472$)   &$1470.1998^{+0.0016}_{-0.0017}$\n        &Mid-transit time of {TOI-530} b.\\\\\n        $r_{1,b}$  &$\\mathcal{U}$ (0\\ ,\\ 1)   &$0.553^{+0.056}_{-0.074}$\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n        $r_{2,b}$  &$\\mathcal{U}$ (0\\ ,\\ 1)   &$0.155^{+0.002}_{-0.002}$\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n        $e_{b}$                     &0  &Fixed  &Orbital eccentricity of {TOI-530} b.\\\\\n        $\\omega_{b}$ (deg)          &90 &Fixed  &Argument of periapsis of {TOI-530} b.\\\\\n       \n       \n        \\it{{\\it TESS}\\ photometry parameters}\\\\\n        $D_{\\rm TESS}$  &Fixed    &$1$      &{\\it TESS}\\ photometric dilution factor.\\\\\n        $M_{\\rm TESS}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$-0.00002^{+0.00009}_{-0.00009}$      &Mean out-of-transit flux of {\\it TESS}\\ photometry.\\\\\n        $\\sigma_{\\rm TESS}$ (ppm) &$\\mathcal{J}$ ($10^{-6}$\\ ,\\ $10^{6}$)  &$0.02^{+10.40}_{-0.01}$\n          &{\\it TESS}\\ additive photometric jitter term.\\\\\n        $q_{1}$       &$\\mathcal{U}$ (0\\ ,\\ 1)           &$0.16^{+0.14}_{-0.09}$   &Quadratic limb darkening coefficient.\\\\\n        $q_{2}$       &$\\mathcal{U}$ (0\\ ,\\ 1)                 &$0.46^{+0.33}_{-0.30}$  &Quadratic limb darkening coefficient.\\\\\n        \n        \\it{El Sauce photometry parameters}\\\\\n        $D_{\\rm el}$  &Fixed    &$1$      &El Sauce photometric dilution factor.\\\\\n        $M_{\\rm el}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)    &$-0.0004^{+0.0008}_{-0.0008}$\n        &Mean out-of-transit flux of El Sauce photometry.\\\\\n        $\\sigma_{\\rm el}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$17.3^{+483.7}_{-16.6}$\n          &El Sauce additive photometric jitter term.\\\\\n        $q_{\\rm el}$   &$\\mathcal{N}$ ($0.66$\\ ,\\ $0.1^{2}$)                &$0.74^{+0.07}_{-0.08}$     &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT2 photometry parameters}\\\\\n        $D_{\\rm MUSCAT2,g}$   &Fixed  &$1$\n          &MUSCAT2 $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,g}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$-0.0016^{+0.0006}_{-0.0005}$\n          &Mean out-of-transit flux of MUSCAT2 $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,g}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$6868.9^{+517.3}_{-515.7}$\n      &MUSCAT2 $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,g}$      &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)            &$0.67^{+0.07}_{-0.07}$        &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,r}$  &Fixed    &$1$\n          &MUSCAT2 $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,r}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0001^{+0.0003}_{-0.0003}$\n       &Mean out-of-transit flux of MUSCAT2 $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,r}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$4667.1^{+219.3}_{-203.6}$\n       &MUSCAT2 $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,r}$     &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)             &$0.66^{+0.05}_{-0.05}$   &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,i}$  &Fixed     &$1$\n          &MUSCAT2 $i$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,i}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0002^{+0.0003}_{-0.0003}$\n          &Mean out-of-transit flux of MUSCAT2 $i$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,i}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$5026.9^{+207.0}_{-193.0}$\n          &MUSCAT2 $i$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,i}$     &$\\mathcal{N}$ ($0.54$\\ ,\\ $0.1^{2}$)             &$0.61^{+0.05}_{-0.05}$   &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,z}$   &Fixed  &$1$\n          &MUSCAT2 $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,z}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)  &$0.0006^{+0.0002}_{-0.0002}$\n          &Mean out-of-transit flux of MUSCAT2 $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,z}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$4504.2^{+144.9}_{-140.6}$\n          &MUSCAT2 $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,z}$      &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)             &$0.58^{+0.05}_{-0.05}$  &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT photometry parameters}\\\\\n        $D_{\\rm MUSCAT,g}$  &Fixed   &$1$\n          &MUSCAT $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,g}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0002^{+0.0009}_{-0.0009}$\n          &Mean out-of-transit flux of MUSCAT $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,g}$ (ppm)  &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$27.6^{+711.3}_{-26.8}$\n          &MUSCAT $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,g}$   &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)              &$0.77^{+0.08}_{-0.08}$    &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,r}$  &Fixed   &$1$\n          &MUSCAT $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,r}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0001^{+0.0003}_{-0.0003}$\n          &Mean out-of-transit flux of MUSCAT $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,r}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)   &$8.96^{+177.7}_{-8.4}$\n          &MUSCAT $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,r}$      &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)          &$0.76^{+0.06}_{-0.05}$     &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,z}$  &Fixed   &$1$\n          &MUSCAT $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,z}$  &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)   &$0.0002^{+0.0002}_{-0.0002}$\n          &Mean out-of-transit flux of MUSCAT $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,z}$ (ppm) &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)  &$11.7^{+192.4}_{-11.1}$\n          &MUSCAT $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,z}$       &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)            &$0.45^{+0.05}_{-0.05}$   &Linear limb darkening coefficient.\\\\\n        \n        \\it{Stellar parameters}\\\\\n        ${\\rho}_{\\ast}$ ($\\rm kg\\ m^{-3}$) &$\\mathcal{J}$ ($100$\\ ,\\ $\\rm 100^{2}$)   &$4278^{+412}_{-395}$\n         &Stellar density.\\\\\n        \\it{RV parameters}\\\\\n        $K_{b}$ ($\\rm m\\ s^{-1}$)  &$\\mathcal{U}$ ($0$\\ ,\\ $200$)      &$66.5^{+14.1}_{-14.0}$\n        &RV semi-amplitude of {TOI-530} b.\\\\\n        $\\rm \\mu_{SPIRou}$ ($\\rm m\\ s^{-1}$) &$\\mathcal{U}$ ($29300$\\ ,\\ $29500$)   &$29402.4^{+11.1}_{-11.5}$\n        &Systemic velocity for SPIRou.\\\\\n        $\\rm \\sigma_{SPIRou}$ ($\\rm m\\ s^{-1}$) &$\\mathcal{J}$ ($0.1$\\ ,\\ $100$)   &$37.3^{+10.8}_{-8.4}$\n        &Extra jitter term for SPIRou.\\\\\n        \\hline\\hline\n    \\label{tranpriors}    \n    \\end{tabular}\n\\end{table*}\n\n\n\\begin{comment}\n\\begin{table*}\n    \\centering\n    \\caption{Prior settings and the best-fit values along with the 68\\% credibility intervals in the final joint fit for {TOI-530}. $\\mathcal{N}$($\\mu\\ ,\\ \\sigma^{2}$) means a normal prior with mean $\\mu$ and standard deviation $\\sigma$. $\\mathcal{U}$(a\\ , \\ b) stands for a uniform prior ranging from a to b. $\\mathcal{J}$(a\\ , \\ b) stands for a Jeffrey's prior ranging from a to b.}\n    \\begin{tabular}{lccr}\n        \\hline\\hline\n        Parameter       &Best-fit Value       &Prior     &Description\\\\\\hline\n        \\it{Planetary parameters}\\\\\n       \n        $P_{b}$ (days)   &$6.387597^{+0.000016}_{-0.000017}$  \n        &$\\mathcal{U}$ ($6.2$\\ ,\\ $6.6$)\n        &Orbital period of {TOI-530} b.\\\\\n        $T_{0,b}$ (BJD-2457000)    &$1470.1999^{+0.0015}_{-0.0016}$ \n        &$\\mathcal{U}$ ($1468$\\ ,\\ $1472$) \n        &Mid-transit time of {TOI-530} b.\\\\\n        $r_{1,b}$    &$0.554^{+0.054}_{-0.045}$ \n        &$\\mathcal{U}$ (0\\ ,\\ 1)\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n        $r_{2,b}$    &$0.155^{+0.001}_{-0.001}$ \n        &$\\mathcal{U}$ (0\\ ,\\ 1)\n        &Parametrisation for {\\it p} and {\\it b}.\\\\\n       \n       \n        $\\mathcal{S}_{1,b}$ &$-0.36^{+0.09}_{-0.07}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        $\\mathcal{S}_{2,b}$ &$0.40^{+0.04}_{-0.04}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        \\it{{\\it TESS}\\ photometry parameters}\\\\\n        $D_{\\rm TESS}$     &$1$ \n        &Fixed      &{\\it TESS}\\ photometric dilution factor.\\\\\n        $M_{\\rm TESS}$    &$-0.00001^{+0.00009}_{-0.00009}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of {\\it TESS}\\ photometry.\\\\\n        $\\sigma_{\\rm TESS}$ (ppm) &$0.02^{+11.88}_{-0.01}$\n        &$\\mathcal{J}$ ($10^{-6}$\\ ,\\ $10^{6}$)      &{\\it TESS}\\ additive photometric jitter term.\\\\\n        $q_{1}$                &$0.16^{+0.14}_{-0.08}$       &$\\mathcal{U}$ (0\\ ,\\ 1)  &Quadratic limb darkening coefficient.\\\\\n        $q_{2}$                &$0.48^{+0.31}_{-0.30}$       &$\\mathcal{U}$ (0\\ ,\\ 1)  &Quadratic limb darkening coefficient.\\\\\n        \n        \\it{El Sauce photometry parameters}\\\\\n        $D_{\\rm el}$     &$1$ \n        &Fixed      &El Sauce photometric dilution factor.\\\\\n        $M_{\\rm el}$    &$-0.0004^{+0.0007}_{-0.0007}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of El Sauce photometry.\\\\\n        $\\sigma_{\\rm el}$ (ppm) &$24.2^{+495.3}_{-23.2}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &El Sauce additive photometric jitter term.\\\\\n        $q_{\\rm el}$                &$0.73^{+0.07}_{-0.07}$       &$\\mathcal{N}$ ($0.66$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT2 photometry parameters}\\\\\n        $D_{\\rm MUSCAT2,g}$     &$1$ \n        &Fixed      &MUSCAT2 $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,g}$    &$-0.0015^{+0.0005}_{-0.0005}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,g}$ (ppm) &$6929.7^{+509.1}_{-497.7}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,g}$                &$0.67^{+0.06}_{-0.05}$       &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,r}$     &$1$ \n        &Fixed      &MUSCAT2 $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,r}$    &$0.0002^{+0.0003}_{-0.0003}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,r}$ (ppm) &$4694.5^{+196.2}_{-186.3}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,r}$                &$0.65^{+0.04}_{-0.05}$       &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,i}$     &$1$ \n        &Fixed      &MUSCAT2 $i$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,i}$    &$0.0002^{+0.0003}_{-0.0003}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $i$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,i}$ (ppm) &$5090.4^{+190.2}_{-184.3}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $i$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,i}$                &$0.60^{+0.05}_{-0.05}$       &$\\mathcal{N}$ ($0.54$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT2,z}$     &$1$ \n        &Fixed      &MUSCAT2 $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT2,z}$    &$0.0005^{+0.0002}_{-0.0002}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT2 $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT2,z}$ (ppm) &$4478.3^{+136.2}_{-135.4}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT2 $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT2,z}$                &$0.58^{+0.04}_{-0.05}$       &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        \\it{MUSCAT photometry parameters}\\\\\n        $D_{\\rm MUSCAT,g}$     &$1$ \n        &Fixed      &MUSCAT $g$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,g}$    &$0.0003^{+0.0008}_{-0.0008}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT $g$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,g}$ (ppm) &$41.8^{+799.1}_{-40.6}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT $g$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,g}$                &$0.76^{+0.07}_{-0.08}$       &$\\mathcal{N}$ ($0.79$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,r}$     &$1$ \n        &Fixed      &MUSCAT $r$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,r}$    &$0.0001^{+0.0003}_{-0.0003}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT $r$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,r}$ (ppm) &$12.6^{+198.2}_{-12.0}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT $r$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,r}$                &$0.75^{+0.06}_{-0.05}$       &$\\mathcal{N}$ ($0.73$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        $D_{\\rm MUSCAT,z}$     &$1$ \n        &Fixed      &MUSCAT $z$ band photometric dilution factor.\\\\\n        $M_{\\rm MUSCAT,z}$    &$0.0002^{+0.0002}_{-0.0002}$\n        &$\\mathcal{N}$ (0\\ ,\\ $0.1^{2}$)      &Mean out-of-transit flux of MUSCAT $z$ band photometry.\\\\\n        $\\sigma_{\\rm MUSCAT,z}$ (ppm) &$7.9^{+144.5}_{-7.4}$\n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $10^{5}$)      &MUSCAT $z$ band additive photometric jitter term.\\\\\n        $q_{\\rm MUSCAT,z}$                &$0.45^{+0.05}_{-0.05}$       &$\\mathcal{N}$ ($0.44$\\ ,\\ $0.1^{2}$)  &Linear limb darkening coefficient.\\\\\n        \n        \\it{Stellar parameters}\\\\\n        ${\\rho}_{\\ast}$ ($\\rm kg\\ m^{-3}$)   &$4506^{+137}_{-144}$\n        &$\\mathcal{J}$ ($100$\\ ,\\ $\\rm 100^{2}$) &Stellar density.\\\\\n        \\it{RV parameters}\\\\\n        $K_{b}$ ($\\rm m\\ s^{-1}$)       &$86.5^{+6.2}_{-6.7}$\n        &$\\mathcal{U}$ ($0$\\ ,\\ $200$)\n        &RV semi-amplitude of {TOI-530} b.\\\\\n        $\\rm \\mu_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$29395.4^{+5.2}_{-5.8}$ \n        &$\\mathcal{U}$ ($29300$\\ ,\\ $29500$)\n        &Systemic velocity for SPIRou.\\\\\n        $\\rm \\sigma_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$5.1^{+6.8}_{-3.2}$ \n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $100$)\n        &Extra jitter term for SPIRou.\\\\\n        \\hline\\hline\n    \\label{tranpriors}    \n    \\end{tabular}\n\\end{table*}\n\\end{comment}\n\n\\begin{table}\n\\caption{Derived physical parameters from the final joint fit for {TOI-530}.}\n\n\n \\begin{tabular}{lcr}   \n    \\hline\\hline\n       \n        Parameter        &Best-fit           &Description\\\\\\hline\n        $R_{p}\/R_{\\ast}$  &$0.155^{+0.002}_{-0.002}$ &Planet radius in units of stellar radius.\\\\\n        $R_{p}$ ($R_{J}$)  &$0.83^{+0.06}_{-0.06}$ &Planet radius.\\\\\n        $M_{P}$ ($M_{J}$)  &$0.40^{+0.09}_{-0.10}$ &Planet mass.\\\\\n        $\\rho_{p}$ ($\\rm g\\ cm^{-3}$)  &$0.93^{+0.49}_{-0.35}$ &Planet density.\\\\\n        ${b}$      &$0.33^{+0.08}_{-0.11}$ &Impact parameter.\\\\\n        $a\/R_{\\ast}$     &$20.97^{+0.65}_{-0.67}$ &Semi-major axis in units of stellar radii.\\\\\n        $a$ (AU)       &$0.052^{+0.005}_{-0.004}$ &Semi-major axis.\\\\\n       \n       \n        $i$ (deg)     &$89.1^{+0.3}_{-0.3}$ &Inclination angle.\\\\\n        $T_{\\rm eq}^{[1]}$ (K)     &$565^{+28}_{-31}$ &Equilibrium temperature.\\\\\n       \n        \\hline\\hline \n    \\end{tabular}\n    \\begin{tablenotes}\n   \n   \n   \n    \\item[1]  [1]\\ We assume there is no heat distribution between the dayside and nightside, and that the albedo is zero.\n    \\end{tablenotes}\n    \\label{physical_parameter}\n\\end{table}\n\n\\begin{comment}\n\\begin{table}\n\\caption{Derived physical parameters from the final joint fit for {TOI-530}.}\n\n\n \\begin{tabular}{lcr}   \n    \\hline\\hline\n       \n        Parameter       &Best Value           &Description\\\\\\hline\n        $R_{p}\/R_{\\ast}$ &$0.155^{+0.001}_{-0.001}$ &Planet radius in units of stellar radius.\\\\\n        $R_{p}$ ($R_{J}$) &$0.83^{+0.05}_{-0.05}$ &Planet radius.\\\\\n        $M_{P}$ ($M_{J}$) &$0.49^{+0.04}_{-0.04}$ &Planet mass.\\\\\n        $\\rho_{p}$ ($\\rm g\\ cm^{-3}$) &$1.14^{+0.35}_{-0.26}$  &Planet density.\\\\\n        ${b}$    &$0.33^{+0.08}_{-0.11}$   &Impact parameter.\\\\\n        $a\/R_{\\ast}$    &$24.85^{+1.62}_{-1.63}$  &Semi-major axis in units of stellar radii.\\\\\n        $a$ (AU)      &$0.062^{+0.008}_{-0.007}$  &Semi-major axis.\\\\\n        $e_{b}$                     &$0.29^{+0.04}_{-0.04}$    &Orbital eccentricity of {TOI-530} b.\\\\\n        $\\omega_{b}$ (deg)          &$7.9^{+17.5}_{-6.4}$  &Argument of periapsis of {TOI-530} b.\\\\\n        $i$ (deg)    &$89.3^{+0.2}_{-0.2}$ &Inclination angle.\\\\\n        $T_{\\rm eq}^{[1]}$ (K)    &$519^{+35}_{-33}$ &Equilibrium temperature.\\\\\n       \n        \\hline\\hline \n    \\end{tabular}\n    \\begin{tablenotes}\n   \n   \n   \n    \\item[1]  [1]\\ We assume there is no heat distribution between the dayside and nightside, and that the albedo is zero.\n    \\end{tablenotes}\n    \\label{physical_parameter}\n\\end{table}\n\\end{comment}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\textwidth]{SPIRou_RV_v2.pdf}\n\\caption{{\\it Left panel}: The systemic velocity-subtracted SPIRou RVs of {TOI-530}\\ as a function of time along with the best-fit circular orbit model from the photometry+RV joint analysis shown as a black solid line. The error bars are the quadrature sum of the instrument jitter term and the measurement uncertainties for all RVs. The orange shaded region represents the $1\\sigma$ confidence interval of the model. {\\it Right panel}: The corresponding phased-folded SPIRou RV data. Residuals are plotted below.} \n\\label{SPIROU}\n\\end{figure*}\n\n\\begin{comment}\n\\begin{table*}\n    \\centering\n    \\caption{Prior settings and the best-fit values along with the 68\\% credibility intervals in the RV modeling for {TOI-530}.}\n    \\begin{tabular}{lccr}\n        \\hline\\hline\n        Parameter       &Best-fit Value       &Prior     &Description\\\\\\hline\n        \\it{Planetary parameters}\\\\\n        $P_{b}$ (days)   &$6.3876$\n        &Fixed\n        &Orbital period of {TOI-530} b.\\\\\n        $T_{0,b}$ (BJD)    &$2458470.199$ \n        &Fixed\n        &Mid-transit time of {TOI-530} b.\\\\\n        $\\mathcal{S}_{1,b}$ &$-0.41^{+0.11}_{-0.10}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        $\\mathcal{S}_{2,b}$ &$0.38^{+0.05}_{-0.07}$ &$\\mathcal{U}$ ($-1$\\ ,\\ $1$) &Parametrisation for $e$ and $\\omega$ of {TOI-530} b. \\\\\n        \n        \\it{RV parameters}\\\\\n        $\\rm \\mu_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$29395.2^{+5.3}_{-5.4}$ \n        &$\\mathcal{U}$ ($29300$\\ ,\\ $29500$)\n        &Systemic velocity for SPIRou.\\\\\n        $\\rm \\sigma_{SPIRou}$ ($\\rm m\\ s^{-1}$)    &$1.2^{+4.7}_{-0.9}$ \n        &$\\mathcal{J}$ ($0.1$\\ ,\\ $100$)\n        &Extra jitter term for SPIRou.\\\\\n\n        $K_{b}$ ($\\rm m\\ s^{-1}$)       &$86.5^{+7.0}_{-6.4}$\n        &$\\mathcal{U}$ ($0$\\ ,\\ $200$)\n        &RV semi-amplitude of {TOI-530} b.\n        \n       \n       \n       \n       \n       \n       \n        \n        \\\\ \\hline\n        \\it{Derived parameters}\\\\ \n        $e_{b}$                     &\\multicolumn{2}{c}{$0.31^{+0.05}_{-0.05}$}   &Orbital eccentricity of {TOI-530} b.\\\\\n        $\\omega_{b}$ (deg)          &\\multicolumn{2}{c}{$9.3^{+23.2}_{-6.9}$}  &Argument of periapsis of {TOI-530} b.\\\\\n        $M_{P}$ ($M_{J}$) &\\multicolumn{2}{c}{$0.49^{+0.04}_{-0.04}$} &Planet mass.\\\\\n        $\\rho_{p}$ ($\\rm g\\ cm^{-3}$) &\\multicolumn{2}{c}{$1.14^{+0.35}_{-0.26}$} &Planet density.\\\\\n         \\hline\\hline \n    \\end{tabular}\n   \n   \n   \n   \n   \n    \\label{rvpriors}\n\\end{table*}\n\\end{comment}\n\n\n\\section{Discussion}\\label{discussion}\n\n\\subsection{A lack of hot massive giant planets around M dwarfs?}\nFigure \\ref{aplot} shows the planet-to-star mass ratio ($q$) as a function of separation distance ($a$) of all giant planets ($0.3\\ M_{J}<M_{p}<13.6\\ M_{J}$) around M dwarfs detected by different methods. Regarding the microlensing sample, since most lens systems are still blended with their sources, it is hard to determine the spectral type of the host star in the lens system\\footnote{Even if the lens and their sources have separated after sufficient long time due to the proper motion, it is still difficult as the host stars of the lens systems are very faint (typically $V\\sim25$ mag).}. Thus we simply set a host-mass threshold between 0.08 and 0.65 $M_{\\odot}$, and we filter out targets that meet the mass cut. While for the other three, we pick out the sample mainly based on the spectral information. We only consider the mass ratio here because most microlensing light-curve analyses do not provide the masses of the host and the planet, although the planet-to-host mass ratio, $q$, is well determined \\citep{Mao1991,Gould1992}. To measure the mass of microlensing planet, one needs two observables \\citep{Zang2020}, but most microlensing planets do not have them, and thus a Bayesian analysis is needed to estimate the host mass, which has a typical $1\\sigma$ uncertainty of $\\sim 0.3~M_{\\odot}$. Thus, it is challenging to classify microlensing planets according to different types of host stars. However, several microlensing detections with unambiguous mass measurements demonstrate that gaint planets orbiting M dwarfs are common (e.g., \\citealt{Bennett2020}).\n\nFour giant planets identified by direct imaging that are far from their host M dwarfs are located at the high-mass-ratio region. This is likely caused by observational biases as the imaging method has difficulty to detect low mass Jupiters with $M_{p}$ around $1\\ M_{J}$ (all of these four planets have $M_{p}\\gtrsim10\\ M_{J}$). Microlensing, however, is sensitive to all kinds of widely separated planets with masses ranging from  super-Jupiter down to Earth (e.g., \\citealt{Zang2021}). A total of 55 giant planets harboured by M dwarfs have been discovered with projected separation distance $a_{\\bot}\\gtrsim1$ AU\\footnote{The solutions of 12 microlensing systems have the so-called close-wide degeneracy, shown in pairs as translucent blue squares in Figure \\ref{aplot}.}. There is a wide mass ratio distribution of those microlensing systems, most of which have $q\\lesssim10^{-2}$, indicating that cold Jupiters around M dwarfs are possibly common and diverse. \n\nA similar trend can also been seen in the RV-only sample whose separation distances are between 0.1 and 10 AU, although RV can only determine the minimum mass ratio $q_{\\rm min}$ for those non-transiting systems. Currently, there are no RV-only giant planets with $q_{\\rm min}\\geq10^{-2}$ that have been detected around M dwarfs, which is likely due to observational biases as follows. Unlike microlensing, which is not limited by the lens flux, determining the companion mass spectroscopically requires central stars to be relatively bright (typically $V<13$ mag). Thus the RV-only sample may miss giant planets around faint late-type M dwarfs, which have higher $q_{\\rm min}$ compared with equivalent planets around early-type M dwarfs. For massive early-type M dwarfs, however, some of their companions within that mass ratio range should belong to brown dwarfs, which are not included here. Furthermore, no giant planets have been detected within 0.1 AU of their host M dwarfs from RV-only surveys. This phenomenon can be attributed to the RV observational strategy. Most RV surveys focus on bright nearby M dwarfs and the total sample size is small (roughly $\\sim200$). Thus it is reasonable to find none RV-only giant planets in this region given the low occurrence rate of hot Jupiter ($\\sim0.5\\%$, \\citealt{Fressin2013,Petigura2018,Zhou2019}). Therefore, deep transit surveys play a crucial role in detecting such candidates as they are sensitive to planets with small semi-major axis and large mass ratio, which also more likely to have a large planet-to-star radius ratio. \n\nInterestingly, all five known transiting giant planet systems and {TOI-530} b turn out to have small mass ratio $q\\sim10^{-3}$ and there is a possible dearth at the region with $q>2\\times10^{-3}$ and $a<0.1$ AU (see the shaded region in Figure \\ref{aplot}). Part of that may result from the flux-limit problem above (for $q\\geq10^{-2}$). We note that this deficiency feature may reflect a more fundamental link to the planet formation theory. Recent work from \\cite{Liu2019} constructed a pebble-driven planet population synthesis model, and their simulation results suggest that gas giants may mainly form when the central stars are more massive than $0.3\\ M_{\\odot}$ (see Figure 7 in \\citealt{Liu2019}). This is because planets stop increasing their core masses when they reach the pebble isolation mass $M_{\\rm iso}$, which is proportional to the stellar mass as $M_{\\rm iso}\\propto M_{\\ast}^{4\/3}$. Following gas accretion onto planets with small $M_{\\rm iso} $ is limited due to a slow Kelvin\u2013Helmholtz contraction. Thus they would stop before the runaway gas accretion and be left as rock- or ice-dominated planets with tiny atmospheres. If this is the case, we then expect few giant planets with relatively high mass ratio above $10^{-3}$ when their host masses are below $0.3\\ M_{\\odot}$. Note that the known ``brown dwarf desert'' ($35 \\leq M\\sin i \\leq 55\\ M_{J}$ and orbital periods under 100 days), studied by \\cite{Ma2014} using all available data of close brown dwarfs around solar-type stars, is also located in this region with $q\\gtrsim0.05$. This lower limit is estimated based on the lower limit of the brown dwarf desert $35\\ M_{J}$ and the typical mass upper limit of M dwarfs $0.65\\ M_{\\odot}$. However, it is still unclear whether the deficiency between $2\\times10^{-3}$ and $10^{-2}$ in mass ratio is physical. Compared with the known planets (red squares plus {TOI-530} b in Figure \\ref{aplot}), the transit method is, in principle, more sensitive to giant planets with a larger mass ratio (i.e., larger radius ratio) located in this deficiency region. Additionally, if detected by transit survey, planet candidates within this parameter space range should be easily confirmed by the RV method. Due to the lack of transiting giant planets around M dwarfs, we cannot draw any conclusions yet. Hopefully, the {\\it TESS}\\ QLP Faint Star ($10.5<T<13.5$ mag) Search could provide more such systems and check if this depletion feature is real (Kunimoto et al, in prep). \n\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{a_q.pdf}\n\\caption{Planet-to-star mass ratio of all giant planets around M dwarfs as a function of semi-major axis. Different colors represent planets detected by different methods. The two horizontal black dashed lines represent the selection threshold of our sample. The upper limit is $q=13.6\\ M_{J}\/0.08\\ M_{\\odot}=0.17$, while the lower limit corresponds to $q=0.3\\ M_{J}\/0.65\\ M_{\\odot}=4.6\\times10^{-4}$. {TOI-530} b is marked as a magenta circle. The grey shaded region represents a possible paucity of hot massive giant planets around M dwarfs.} \n\\label{aplot}\n\\end{figure}\n\n\\subsection{Metallicity Dependence}\\label{metallicity_dependence}\nAlthough the core accretion theory \\citep{Pollack1996} has predicted rare giant planets around M dwarfs due to their low protoplanetary disk mass as $M_{\\rm disk}$ linearly scales with the stellar mass $M_{\\ast}$ \\citep{Andrews2013}, this defect may be compensated if the parent star is metal rich, which could theoretically supply more solid material used for accretion. Alternatively, gravitational instability (GI) could also form giant planets around M dwarfs \\citep{Boss2006}, though simulation work from \\cite{Cai2006} suggested that GI is unlikely the major mechanism that produce most observed planets. Under the GI hypothesis, we expect that there would not exist a strong dependence between giant planet formation and host metallicity and that even stars with relatively low metallicity should harbour gas giants \\citep{Boss2002}.\n\nIn order to investigate the metallicity dependence observationally, we retrieve a list of solar-type stars (simply selected based on $0.90\\ M_{\\odot}<M_{\\ast}<1.06\\ M_{\\odot}$) hosting giant planets from the NASA Exoplanet Archive \\citep{Akeson2013}. We find a total of 88 transiting and 102 RV-only systems. Most transiting giant planets are hot with semi-major axis $a\\lesssim 0.1$ AU, while the majority of RV-only planets are cold with semi-major axes beyond  $0.1$ AU. Figure~\\ref{fehplot} illustrates the metallicity distribution of their hosts, indicating that hot and cold giant planets do not present much difference in [Fe\/H] preference around solar-type stars (see the transparent points in Figure \\ref{fehplot}). The weighted-mean metallicity of both transiting and RV-only giant planet central stars is above the solar value but almost the same: 0.12 and 0.14, respectively. This is consistent with the conclusions from early work suggesting that the frequency of giant planets increases with stellar metallicity \\citep{Santos2004,Fischer2005,Sousa2011}. However, it is not the same for gas giants around M dwarfs. \n\nAmong all giant planets around M dwarfs, we only find 4 transiting and 9 RV-only systems that have metallicity measurements in the literature. For cold (RV-only) giant planets, the metallicities of their M-dwarf hosts are distributed on both sides of the median value 0.14 (the green dashed line in Figure \\ref{fehplot}) seen for the solar-type stars, though the uncertainties are large. Some of them are formed around metal poor M stars (e.g., GJ 832b, \\citealt{Bailey2009}). This is plausibly in agreement with the previous finding that clump formation is fairly insensitive to the metallicity of the parent star \\citep{Boss2002}, implying that part of the formation of cold giant planets around M dwarfs may take place through GI. Indeed, the recently detected planetary system GJ 3512b, whose host star has a solar metallicity $-0.07\\pm0.16$, favours the GI formation scenario \\citep{Morales2019}.\n\nHowever, the host stars of four known transiting giant planets together with {TOI-530}\\ tend to be metal rich, all of which have [Fe\/H] higher than the aforementioned median value 0.12 (the red dashed line in Figure \\ref{fehplot}). It indicates that the formation of hot (transiting) giant planets around M dwarfs may have a strong dependence on the host metallicity as predicted by the core accretion theory. This is consistent with the recent findings from \\cite{Maldonado2020} (see the left panel of their Figure 14), which reveals a correlation between the metallicities of M dwarfs and their probability of hosting giant planets. Compared with hot giant planets around solar-type stars, the formation of hot giant planets around M dwarfs possibly requires higher host metallicity, though the number of detections is probably too small to confirm this claim. To examine the statistical significance of this feature, we carry out a Kolmogorov-Smirnov (K-S) test. We calculate the K-S statistic between the metallicities of stars in the G- and M-type transiting sample, which yields a $p$ value of 0.023. It roughly corresponds to the $2.5\\sigma$ significance level, at which we can reject the null hypothesis that two samples are from the same distribution. We then adopt the bootstrap method to randomly draw distributions from the G-type parent sample and compute the K-S statistic between these random distributions and the M-type transiting sample. We repeat this procedure for 10,000 times and derive the corresponding $p$ value distribution. We find that 88\\% of the total random samples have $p$ values less than 0.05 ($2\\sigma$ level) while only 0.3\\% of them have $p$ values less than 0.003 ($3\\sigma$ level), indicating a marginal correlation. Future detections of more hot giant planets around M dwarfs will reveal whether this metallicity preference is significant or not.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{feh_M_G.pdf}\n\\caption{Planet mass $M_{p}$ vs. host star metallicity. Different colors represent planets detected by different methods (transit\/RV). The red and blue dashed lines are the weighted-mean metallicity of solar-type stars hosting giant planets detected by transit and RV-only observations. {TOI-530} b is marked as a magenta circle. All hot giant planet M-dwarf hosts tend to have a higher metallicity than G dwarf hosts (see Section \\ref{metallicity_dependence}).} \n\\label{fehplot}\n\\end{figure}\n\n\n\n\n\\begin{comment}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{P-e.pdf}\n\\caption{Orbital eccentricity of all giant planets around M dwarfs as a function of planet orbital period. Different color represent planets detected by different methods. The grey points mark confirmed giant planets around other types of stars with non-zero eccentricity measurements (uncertainty smaller than 30\n\\%). Data are retrieved from NASA Exoplanet Archive \\citep{Akeson2013}. {TOI-530} b is marked as a magenta circle.} \n\\label{peplot}\n\\end{figure}\n\\end{comment}\n\n\n\\subsection{Prospects for future observations}\nDue to the faintness of the host star, high precision radial velocity spectrographs on large telescopes like the red-optical MAROON-X \\citep{Seifahrt2018} or near-infrared instruments like InfrRed Doppler spectrograph (IRD; \\citealt{Kotani2018}) are required to reach sufficient signal-to-noise ratio. The host star is quiet without strong intrinsic stellar activity as there are no significant flux variations in the {\\it TESS}\\ light curve, making it a suitable target for precise RV follow-up observations. Though a potential 9.4 d modulation signal is shown up in the {\\it TESS}\\ light curve, this is probably not linked to the stellar rotation given our ZTF results and previous findings from \\cite{Newton2018} (see Section \\ref{stellar_properties}).\n\nMeasuring the stellar obliquity of M dwarfs hosting hot giant planets could provide clues about their origins and gain insights into their migration history. To probe the potential opportunities to observe the Rossiter-McLaughlin effect \\citep{Rossiter1924,McLaughlin1924} of {TOI-530}\\ and measure the projected\nangle between the planet orbital and stellar equatorial planes, we estimate the RM semi-amplitude of this system as:\n\\begin{equation}\n    A_{\\rm RM} \\simeq \\frac{2}{3}(R_{p}\/R_{\\ast})^{2}\\sqrt{1-b^{2}}\\times v \\sin i,\n\\end{equation}\nwhere $b$ is the impact parameter and $v \\sin i$ is the projected stellar equatorial rotation velocity. Taking the best-fit values from the light curve modeling, assuming a rotational period of 40 d for a $0.5\\ M_{\\odot}$ star (see Figure 4 in \\citealt{Newton2018}) with an upper and lower limit of 100 and 30 d, we find $A_{\\rm RM}\\sim 10^{+4}_{-6}$ m\/s, making the RM signal detectable with near-infrared RV observations.\n\nFinally, we investigate the atmospheric characterization possibility by calculating the Transmission Spectroscopy Metric (TSM; \\citealt{Kempton2018}) of {TOI-530} b. We obtain a TSM of $41^{+20}_{-13}$, which is much smaller than the recommended threshold of 90 for high-quality sub-Jovians ($4.0<R_{p}<10\\ R_{\\oplus}$) targets from \\cite{Kempton2018}. Thus we conclude that {TOI-530} b is not a promising target for future atmospheric composition studies. \n\n\n\\section{Summary and Conclusions}\\label{conclusion}\nIn this paper, we present the discovery and characterization of a transiting giant planet {TOI-530} b around an early, metal-rich M dwarf. Space and ground photometry as well as SPIRou RVs reveal that {TOI-530} b has a radius of $0.83\\pm0.05\\ R_{J}$ and a mass of $0.4\\pm0.1\\ M_{J}$ on a 6.39-d orbit. Although it is challenging to probe the atmospheric properties of {TOI-530} b due to the faintness of the host star, {TOI-530}\\ is still a suitable target to study the stellar obliquity. Furthermore, we report a potential paucity of hot massive giant planets around M dwarfs with separation distance smaller than $0.1$ AU and planet-to-star mass ratio between $2\\times10^{-3}$ and $10^{-2}$. We also identify a possible correlation between hot giant planet formation and the metallicity of its parent M dwarf. However, due to the current small sample of such systems, we could not make any firm conclusions in this context. Future near-infrared spectroscopic surveys such as SPIRou Legacy Survey-Planet Search (SLS-PS; \\citealt{Moutou2017,Fouque2018}) shall remedy this situation.\n\n\\section*{Affiliations}\n$^{1}$Department of Astronomy, Tsinghua University, Beijing 100084, People's Republic of China\\\\\n$^{2}$Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China\\\\\n$^{3}$National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, People's Republic of China\\\\\n$^{4}$Canada-France-Hawaii Telescope, CNRS, Kamuela, HI 96743, USA\\\\\n$^{5}$Univ. de Toulouse, CNRS, IRAP, 14 Avenue Belin, 31400 Toulouse, France\\\\\n$^{6}$Department of Physics and Astronomy, Vanderbilt University, 6301 Stevenson Center Ln., Nashville, TN 37235, USA\\\\\n$^{7}$Department of Physics, Fisk University, 1000 17th Avenue North, Nashville, TN 37208, USA\\\\\n$^{8}$Department of Astronomy, University of California Berkeley, Berkeley, CA 94720, USA\\\\\n$^{9}$Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\\\\\n$^{10}$Instituto de Astrof\\'isica de Canarias (IAC), V\\'ia L\\'actea s\/n, E-38205 La Laguna, Tenerife, Spain\\\\\n$^{11}$Dept. Astrof\\'sica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain\\\\\n$^{12}$NASA Exoplanet Science Institute, Caltech\/IPAC, Mail Code 100-22, 1200 E. California Blvd., Pasadena, CA 91125, USA\\\\\n$^{13}$NASA Ames Research Center, Moffett Field, CA 94035, USA\\\\\n$^{14}$Center for Astrophysics ${\\rm \\mid}$ Harvard {\\rm \\&} Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA\\\\\n$^{15}$Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\\\\\n$^{16}$NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA\\\\\n$^{17}$University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA\\\\\n$^{18}$Department of Physics \\& Astronomy, University of Kansas, 1082 Malott,1251 Wescoe Hall Dr., Lawrence, KS 66045, USA\\\\\n$^{19}$Embry-Riddle Aeronautical University, Prescott, AZ\\\\\n$^{20}$El Sauce Observatory, Coquimbo Province, Chile\\\\\n$^{21}$Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\\\\\n$^{22}$Department of Astronomy, California Institute of Technology, Pasadena, CA 91125, USA\\\\\n$^{23}$Okayama Observatory, Kyoto University, 3037-5 Honjo, Kamogatacho, Asakuchi, Okayama 719-0232, Japan\\\\\n$^{24}$Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\\\\\n$^{25}$Department of Earth and Planetary Science, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\\\\\n$^{26}$Zhejiang Institute of Modern Physics, Department of Physics \\& Zhejiang University-Purple Mountain Observatory Joint Research Center for Astronomy, Zhejiang University, Hangzhou 310027, China\\\\\n$^{27}$Department of Astronomy, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan\\\\\n$^{28}$U.S. Naval Observatory, Washington, D.C. 20392, USA\\\\\n$^{29}$Japan Science and Technology Agency, PRESTO, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan\\\\\n$^{30}$Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan\\\\\n$^{31}$Department of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA\\\\\n$^{32}$Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA\\\\\n$^{33}$Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA\\\\\n$^{34}$Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA\\\\\n$^{\\ast}$51 Pegasi b Fellow\\\\\n\\section*{Acknowledgements}\nWe are grateful to Coel Hellier for the insights regarding the WASP data. We thank Elisabeth Newton, Robert Wells, Hongjing Yang and Weicheng Zang for useful discussions. We also thank Elise Furlan for the contributions to the speckle data and Nadine Manset for scheduling the SPIRou observations.\nThis work is partly supported by the National Science Foundation of China (Grant No. 11390372 and 11761131004 to SM and TG). \nThis research uses data obtained through the Telescope Access Program (TAP), which has been funded by the TAP member institutes.\nThis work is partly supported by JSPS KAKENHI Grant Numbers JP17H04574 , JP18H05439, 20K14521, JST PRESTO Grant Number JPMJPR1775, and the Astrobiology Center of National Institutes of Natural Sciences (NINS) (Grant Number AB031010).\nThis article is based on observations made with the MuSCAT2 instrument, developed by ABC, at Telescopio Carlos S\u00e1nchez operated on the island of Tenerife by the IAC in the Spanish Observatorio del Teide.\nSome of the observations in the paper made use of the High-Resolution Imaging instrument 'Alopeke obtained under LLP GN-2021A-LP-105. 'Alopeke was funded by the NASA Exoplanet Exploration Program and built at the NASA Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett Quigley. Data were reduced using a software pipeline originally written by Elliott Horch and Mark Everett. 'Alopeke was mounted on the Gemini North telescope of the international Gemini Observatory, a program of NSF's OIR Lab, which is managed by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. on behalf of the Gemini partnership: the National Science Foundation (United States), National Research Council (Canada), Agencia Nacional de Investigaci\u00f3n y Desarrollo (Chile), Ministerio de Ciencia, Tecnolog\u00eda e Innovaci\u00f3n (Argentina), Minist\u00e9rio da Ci\u00eancia, Tecnologia, Inova\u00e7\u00f5es e Comunica\u00e7\u00f5es (Brazil), and Korea Astronomy and Space Science Institute (Republic of Korea). \nFunding for the TESS mission is provided by NASA's Science Mission directorate. \nWe acknowledge the use of {\\it TESS}\\ public data from pipelines at the {\\it TESS}\\ Science Office and at the {\\it TESS}\\ Science Processing Operations Center. \nResources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the production of the SPOC data products.\nThis research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program. \nThis paper includes data collected by the {\\it TESS}\\ mission, which are publicly available from the Mikulski Archive for Space Telescopes\\ (MAST). \nThis work has made use of data from the European Space Agency (ESA) mission\n{\\it Gaia} (\\url{https:\/\/www.cosmos.esa.int\/gaia}), processed by the {\\it Gaia} Data Processing and Analysis Consortium (DPAC,\n\\url{https:\/\/www.cosmos.esa.int\/web\/gaia\/dpac\/consortium}). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the {\\it Gaia} Multilateral Agreement.\nThis work made use of \\texttt{tpfplotter} by J. Lillo-Box (publicly available in www.github.com\/jlillo\/tpfplotter), which also made use of the python packages \\texttt{astropy}, \\texttt{lightkurve}, \\texttt{matplotlib} and \\texttt{numpy}.\n\\section*{Data Availability}\nThis paper includes photometric data collected by the {\\it TESS}\\ mission and ground instruments, which are publicly available in ExoFOP, at \\url{https:\/\/exofop.ipac.caltech.edu\/tess\/target.php?id=387690507}. All spectroscopy data underlying this article are listed in the text. All of the high-resolution speckle imaging data is available at the NASA exoplanet Archive with no proprietary period.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Divisors on smooth manifolds}\n\\label{sec:divonmanifolds}\nIn this section we introduce real and complex divisors on smooth manifolds (see also \\cite{CavalcantiKlaasse18,Klaasse17} and \\cite{CavalcantiGualtieri18, VanderLeerDuran16}). These are extensions of the complex geometric notion of divisors to the smooth setting. The theory of divisors permeates much of this paper, as they will be used to keep track of degeneracies of both Lie algebroids and Poisson structures. Let $X$ be a fixed smooth manifold and let $\\mathcal{C}^\\infty_X$ be its sheaf of smooth functions, with $C^\\infty(X) = \\mathcal{C}^\\infty_X(X)$.\n\nIn this section we will often not specify whether we are dealing with real or complex objects, and treat them simultaneously and on equal footing. However, in the remainder of the paper we will mostly consider real divisors. Their complex counterparts will play a larger role in the follow-up paper \\cite{Klaasse18} (and occur prominently in \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18,VanderLeerDuran16}).\n\\begin{defn} A \\emph{divisor} $(U,\\sigma)$ on $X$ is a line bundle $U \\to X$ together with a section $\\sigma \\in \\Gamma(U)$ with nowhere dense closed zero set $Z_\\sigma := \\sigma^{-1}(0) \\subseteq X$. A \\emph{divisor ideal sheaf} is a locally principal ideal sheaf $\\mathcal{I} \\subseteq \\mathcal{C}^\\infty_X$ generated by nowhere-densely vanishing functions. Its associated \\emph{divisor ideal} is the ideal of its global sections, $I := \\mathcal{I}(X) \\subseteq C^\\infty(X)$.\n\\end{defn}\n\\begin{rem} Recall that a nowhere dense subset of a topological space is one whose closure has empty interior. In other words, its intersection with any nonempty open subset is not dense. Often the subset $Z_\\sigma$ will be a (union of) submanifold(s) of positive codimension.\n\\end{rem}\nAny divisor specifies a divisor ideal $I_\\sigma := \\sigma(\\Gamma(U^*)) \\subseteq C^\\infty(X)$ by evaluation, with associated divisor ideal sheaf $\\mathcal{I}_\\sigma$. Letting $\\alpha$ be a local trivialization of $U^*$, we have $\\alpha(\\sigma) = g$ for some local function $g$, and hence locally $I_\\sigma = \\langle \\alpha(\\sigma) \\rangle = \\langle g \\rangle$. Moreover, note that\n\\begin{equation*}\n\tZ(I) := \\{x \\in X \\, | \\, f(x) = 0 \\text{ for all } f \\in \\mathcal{I}_x\\}\n\\end{equation*}\nis the support of the quotient sheaf $\\mathcal{C}^\\infty_X \/ \\mathcal{I}$, with $Z(I_\\sigma) = Z_\\sigma$ (here $\\mathcal{I}_x$ denotes the stalk of $\\mathcal{I}$ at $x \\in X$). Because of this we will say that $(U,\\sigma)$ is \\emph{supported} on $Z_\\sigma$, and sometimes write $Z_I := Z(I)$. Similarly we say that the divisor ideal $I$ is supported on $Z_I \\subseteq X$.\n\nMorphisms of divisors are most conveniently described through their associated divisor ideals. A \\emph{morphism of divisors} $(U,\\sigma) \\to X$ and $(U',\\sigma') \\to X'$ is a map $f\\colon X \\to X'$ satisfying $f^* I_{\\sigma'} = I_\\sigma$, where $f^* I \\subseteq C^\\infty(X)$ for an ideal $I \\subseteq C^\\infty(X')$ denotes the ideal generated by the pullback. Divisors are \\emph{diffeomorphic} if $f$ is a diffeomorphism, while they are \\emph{isomorphic} (written using $=$) if $X = X'$ and the identity map ${\\rm id}_X$ is a morphism of divisors.\n \nA divisor can be reconstructed up to isomorphism by its divisor ideal. This establishes a bijective correspondence between divisor ideals and isomorphism classes of divisors. This allows us to mostly work with divisor ideals, which carry all essential information. However, often there is a canonical representative of its divisor isomorphism class, for example for Lie algebroid morphisms and Poisson structures.\n\\begin{prop}[\\cite{CavalcantiKlaasse18,VanderLeerDuran16}]\\label{prop:locprincideal} Let $I \\subseteq C^\\infty(X)$ be a divisor ideal. Then there exists a divisor $(U_I,\\sigma)$ on $X$ unique up to isomorphism such that $I_\\sigma = I$.\n\\end{prop}\nDivisors over a fixed manifold $X$ admit a \\emph{product} operation, given by the tensor product $(U,\\sigma) \\otimes (U',\\sigma') := (U \\otimes U', \\sigma \\otimes \\sigma')$. The \\emph{trivial divisor} $(\\underline{\\mathbb{R}},\\underline{1})$ is the unit up to isomorphism. This operation produces another divisor, because the union of nowhere dense subsets is nowhere dense. The product of divisors corresponds to the product of divisor ideals, i.e.\\ $I_{\\sigma \\otimes \\sigma'} = I_\\sigma \\cdot I_{\\sigma'}$, and hence descends to isomorphism classes, with unit $C^\\infty(X)$, the trivial divisor ideal. A divisor ideal $I$ is \\emph{divisible} by $I'$ if $I = I' \\cdot I''$ for some divisor ideal $I''$, and similarly at the level of divisors. We will write $I' < I$ if the divisor ideal $I$ is divisible by $I'$.\n\\begin{rem} Divisors ideals on $X$ with this notion of morphism and product form an abelian category $\\mathfrak{Div}(X)$. Accordingly, we can consider (ir)reducible elements. This leads to asking whether one is able to categorize or classify irreducible divisor ideals. This is akin to the classification of singularities, and related to standardness (\\autoref{defn:divprojstandard}).\n\\end{rem}\n\\begin{defn}\\label{defn:smoothdivisor} A divisor ideal $I \\subseteq C^\\infty(X)$ is \\emph{smooth} if $Z_I \\subseteq X$ is a smooth submanifold.\n\\end{defn}\nHere in the notion of submanifold we allow the codimensions of connected components to vary. Any smooth divisor can be uniquely written as the product of divisors on the connected components of its support, thus we can sometimes assume that the support is connected.\n\\begin{prop}\\label{prop:divvanishingideal} Let $(U,\\sigma)$ be a divisor. Then $I_\\sigma \\subseteq I_{Z_\\sigma}$, the vanishing ideal of $Z_\\sigma$. \n\\end{prop}\n\\bp Locally trivialize $U^*$ by $\\alpha^*$ and let $g := \\alpha(\\sigma) \\in \\mathcal{C}^\\infty_X$. Then locally $I_\\sigma = \\sigma(\\Gamma(U^*)) = \\langle g \\rangle$. As $\\sigma(Z_\\sigma) = 0$, we conclude that $g(Z_\\sigma) = \\sigma(Z_\\sigma) = 0$ so that $g \\in I_{Z_\\sigma}$, hence $I_\\sigma \\subseteq I_{Z_\\sigma}$. \n\\end{proof}\nNote that $I_\\sigma = I_{Z_\\sigma}$ is only possible for $Z_\\sigma$ smooth if $Z_\\sigma$ is of codimension one, as can be seen by counting local generators.\nThere is also a relation between morphisms of divisors and their underlying zero sets. We first introduce the following terminology (c.f.\\ \\cite{CavalcantiKlaasse18}).\n\\begin{defn}\\label{defn:pairsandmaps} A \\emph{pair} $(X,Z)$ is a manifold $X$ with a subset $Z \\subseteq X$. A \\emph{map of pairs} $f\\colon (X,Z) \\to (X',Z')$ is a smooth map $f\\colon X \\to X'$ for which $f(Z) \\subseteq Z'$. A \\emph{strong map of pairs} is a map of pairs $f\\colon (X,Z) \\to (X,Z')$ for which $f^{-1}(Z') = Z$. A map of pairs $f\\colon (X,Z) \\to (X',Z')$ where $Z' \\subseteq X'$ is a submanifold is \\emph{transverse} if $f$ is transverse to $Z'$.\n\\end{defn}\n\\begin{prop}\\label{prop:divmorphstrong} Let $f\\colon (X,U,\\sigma) \\to (X',U',\\sigma')$ be a morphism of divisors. Then $f\\colon (X,Z_{\\sigma}) \\to (X',Z_{\\sigma'})$ is a strong map of pairs, equivalently, satisfies the morphism condition $f^* I_{Z_{\\sigma'}} = I_{Z_{\\sigma}}$.\n\\end{prop}\nIn general, for a map $f\\colon X \\to Y$ and $Z \\subseteq Y$ closed, one can only conclude that $f^* I_Z \\subseteq I_{f^{-1}(Z)}$.\n\\bp As $f$ is a morphism of divisors, we have that $U \\cong f^* U'$ and $\\sigma = g f^* \\sigma'$ for a nonvanishing function $g$. As $Z_{\\sigma}$ is by definition the zero set of $\\sigma$, it is immediate that it equals $f^{-1}(Z_{\\sigma'})$.\n\\end{proof}\n\nGiven a smooth map $f\\colon X \\to X'$ and a divisor $(U',\\sigma')$ on $X'$, one can equip $X$ with a divisor by setting $(U,\\sigma) := (f^* U', f^* \\sigma')$, as long as $f^{-1}(Z_{\\sigma'})$ is nowhere dense (e.g.\\ by assuming $f$ is a transverse map of pairs onto $(X,Z_{\\sigma'})$ if $Z_{\\sigma'}$ is smooth). This automatically makes $f$ a morphism of divisors. If $f$ is transverse to $Z_{\\sigma'}$, the divisor-type is preserved under pullback: by this we mean that the type of degeneracy is similar, so that the class of divisor is preserved.\n\nDivisors admit another type of product. Let $(X,U,\\sigma)$ and $(X',U',\\sigma')$ be two divisors and consider the product $X \\times X'$ with projections $p_{X}\\colon X \\times X' \\to X$ and $p_{X'}\\colon X \\times X' \\to X'$.\n\\begin{defn}\\label{defn:divdirectstum} The \\emph{external tensor product} of $(U,\\sigma)$ and $(U',\\sigma')$ is the divisor\n\\begin{equation*}\n\t(p_X^* U \\otimes p_{X'}^* U', p_X^* \\sigma \\otimes p_{X'}^* \\sigma') \\to X \\times X',\n\\end{equation*}\nwith zero set $Z_\\sigma \\times X' \\cup X \\times Z_{\\sigma'}$ and divisor ideal given by the product $p_X^* I_\\sigma \\cdot p_{X'}^* I_{\\sigma'} \\subseteq C^\\infty(X \\times X')$.\n\\end{defn}\n\\subsection{Examples}\n\\label{sec:divexamples}\nDivisors of various flavors exist, and can naturally be combined using the product operation and external tensor product to form more involved examples. Here we give several examples.\n\\begin{exa}[Trivial divisor] Let $U$ be the trivial line bundle with $\\sigma \\in \\Gamma(U)$ nonvanishing. Then $Z_\\sigma$ is empty, and $(U,\\sigma)$ is called the \\emph{trivial divisor} on $X$. In this case, $I_\\sigma = C^\\infty(X)$.\n\\end{exa}\n\\begin{exa}[Log divisors \\cite{CavalcantiKlaasse18}]\\label{exa:log} A \\emph{log divisor} is a divisor for which $\\sigma$ is transverse to the zero section in $\\Gamma(U)$. Its zero set $Z = Z_\\sigma$ is a smooth hypersurface, and the corresponding divisor ideal is the vanishing ideal $I_Z$, locally generated around $Z$ by $\\langle z \\rangle$, with $z$ a local defining function for $Z$. As shown in \\cite{Klaasse17}, this means that $Z$ specifies a unique isomorphism class of log divisors with zero set $Z$. We will henceforth denote log divisors by $Z = (L_Z,s)$.\n\\end{exa}\n\\begin{exa}[Normal-crossing log divisors \\cite{GualtieriLiPelayoRatiu17}]\\label{exa:normalcrossinglog} The above example can be extended to normal-crossing log divisors $\\underline{Z} := \\bigcup_j Z_j$ associated to a collection $\\{Z_j\\}$ of hypersurfaces which intersect transversely. Such a divisor is the product of the individual log divisors for each $Z_j$, hence locally $I_{\\underline{Z}} = \\langle z_1 \\cdot \\ldots \\cdot z_n \\rangle$. These specify hyperbolic singularities on two-fold intersections: there we can readily change coordinates so that $I_{\\underline{Z}} = \\langle z_1 z_2 \\rangle = \\langle z'^2_1 - z'^2_2 \\rangle$. There is also an extension of these ideas to immersed hypersurfaces with transverse self-crossings \\cite{MirandaScott18}.\n\\end{exa}\nThere is another example resembling \\autoref{exa:normalcrossinglog}, namely that of a \\emph{star log divisor} \\cite{Lanius17}, consisting of the product of log divisors associated to a collection $\\{Z_j\\}$ of pairwise transversely intersecting hypersurfaces $Z_j \\subseteq X$.\n\\begin{exa}[Elliptic divisors \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}]\\label{exa:elliptic} An \\emph{elliptic divisor} is one for which $\\sigma$ vanishes along a smooth codimension-two critical submanifold along which its normal Hessian\n\\begin{equation*}\n\t{\\rm Hess}(\\sigma) \\in \\Gamma(Z_\\sigma; {\\rm Sym}^2 NZ_\\sigma \\otimes U)\n\\end{equation*}\nis definite. We will denote $Z_\\sigma$ by $D$ and the elliptic divisor by $|D| = (R,q)$. As shown in \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}, the associated divisor ideal $I_{|D|} := I_q$ is locally generated by $\\langle r^2 \\rangle$ in polar coordinates $(r,\\theta)$ normal to $D$. Note that $I_{|D|}$ is not the vanishing ideal $I_D$ of $D$.\n\\end{exa}\n\\begin{rem} Hypersurfaces carry a unique log divisor ideal structure. This does not hold for codimension-two submanifolds and elliptic divisors. A simple example is provided by $X = \\mathbb{R}^2$ with $D = \\{(0,0)\\}$ and coordinates $(x,y)$. Equip $D$ with the elliptic ideals $I = \\langle x^2 + y^2 \\rangle$ and $I' = \\langle x^2 + 2y^2 \\rangle$. As these ideals are distinct, they supply $D$ with two non-isomorphic elliptic divisor structures. However, it is easy to see that these elliptic divisors are diffeomorphic.\n\\end{rem}\nWe construct a more elaborate class of divisors out of log and elliptic divisors as follows.\n\\begin{exa}[Elliptic-log divisors]\\label{exa:ellipticlog} An \\emph{elliptic-log divisor} is a divisor obtained as the product of a log divisor $Z = (L_Z,s)$ and an elliptic divisor $|D| = (R,q)$ such that $D \\subseteq Z$ (we will write $|D| \\subseteq Z$ as a shorthand for this). We denote this product by $W := Z \\otimes |D|$, with divisor ideal $I_W = I_Z \\cdot I_{|D|}$. As such, pointwise around $D$ there exist normal coordinates $(x,y)$ to $D$ with $\\{x = 0\\}$ cutting out $Z$, for which $I_W = \\langle x (x^2 + y^2) \\rangle$, with $r^2 = x^2 + y^2$.\n\\end{exa}\nIt is also possible for a log divisor $Z$ and an elliptic divisor $|D|$ to either not or only partially intersect. In general, one can readily combine divisors with nowhere intersecting supports using the product operation. However, most interesting are cases where the divisors do intersect (yet retain interesting properties), of which \\autoref{exa:ellipticlog} is a specific instance.\n\\begin{exa}[Complex log divisors \\cite{CavalcantiGualtieri18}]\\label{exa:complexlog} A \\emph{complex log divisor} is a complex divisor $(U,\\sigma)$ for which $\\sigma$ is transverse to zero. This is the complex analogue of \\autoref{exa:log}, and locally we have $I_\\sigma = \\langle w \\rangle$ for $w$ a local complex function cutting out $D$. Its zero set $D$ is smooth of codimension two, and by complex conjugation we obtain another complex log divisor $(\\overline{U},\\overline{\\sigma})$ with $I_{\\overline{\\sigma}} = \\langle \\overline{w} \\rangle$. After taking their product we can extract an elliptic divisor $((U \\otimes \\overline{U})_\\mathbb{R},\\sigma \\otimes \\overline{\\sigma})$ with $I_{\\sigma \\otimes \\overline{\\sigma}} = \\langle w \\overline{w} \\rangle = \\langle r^2 \\rangle$. Given an elliptic divisor and a choice of coorientation for $D$, there is a complex log divisor inducing it which is unique up to diffeomorphism.\n\\end{exa}\nThere is a general class of divisors of \\emph{Morse--Bott type}, i.e.\\ those pairs $(U,\\sigma)$ for which $\\sigma$ is a \\emph{Morse--Bott section}. This means that $Z_\\sigma$ consists of nondegenerate critical submanifolds, i.e.\\ $Z_\\sigma$ is a union of submanifolds $S$ of $\\sigma$, where $\\ker {\\rm Hess}(\\sigma)_p = T_p S$ for all $p \\in S$. If $U$ is trivial, then a Morse--Bott section for $U$ is not the same thing as a Morse--Bott function on $X$, as we only demand a condition on its zero set. In this sense, an alternate way of phrasing the definition is that $\\sigma$ must specify the germ of a Morse--Bott function around $Z_\\sigma$.\n\nThe elliptic divisors of \\autoref{exa:elliptic} are of Morse--Bott type, but normal-crossing log divisors (\\autoref{exa:normalcrossinglog}) are not. The analogue of the linearization result for elliptic divisors (see \\cite[Proposition 2.20]{CavalcantiKlaasse18}) is readily seen to hold for all divisors of Morse--Bott type (see \\cite{Klaasse17}), so that locally the associated divisor ideal $I_\\sigma$ will be given by $\\langle Q_g \\rangle$ with $Q_g$ a homogeneous function in normal bundle coordinates. Note that all Morse--Bott type divisors are smooth.\n\\begin{rem} Elliptic divisors arise as the real divisors which underlie complex log divisors (c.f.\\ \\cite{CavalcantiGualtieri18}). There is another as of yet unexplored class of real divisors, namely those underlying \\emph{quaternionic log divisors}, i.e.\\ coming from a quaternionic line bundle with a transversally vanishing section. These are supported on smooth submanifolds of codimension four, and will be explored further in future work.\n\\end{rem}\n\n\\section{Introduction}\n\\label{sec:introduction}\nPoisson geometry can be viewed as the study of singular symplectic foliations. Poisson structures range from the zero Poisson structure, whose symplectic foliation is by points, to nondegenerate Poisson structures having only one symplectic leaf, the entire manifold. Within this wide range of possibilities, one can impose restrictions on the characteristics of the symplectic foliations that may occur, to obtain subclasses of Poisson manifolds.\n\nIn this paper we focus on Poisson structures whose symplectic foliation is relatively mild. These are the Poisson structures of divisor-type: those Poisson structures which are generically regular, and whose degeneracy is governed by a divisor ideal. These divisor ideals are locally principal ideals whose local generators have nowhere dense zero set, which we will elaborate on below. Examples of such Poisson structures are those which are generically nondegenerate, i.e.\\ those whose union of open symplectic leaves is dense.\n\nThe class of Poisson structures of divisor-type coincides with the almost-regular Poisson structures of Androulidakis--Zambon \\cite{AndroulidakisZambon17}. They define almost-regular Poisson structures from the point of view of singular foliations, namely as those Poisson structures $\\pi$ on a manifold $X$ whose module of Hamiltonian vector fields $\\mathcal{F}_\\pi = \\pi^\\sharp(\\Gamma(T^*X)) \\subseteq \\Gamma(TX)$ is projective. By results of \\cite{Debord01}, this class is characterized by having a smooth associated holonomy groupoid.\n\nOur viewpoint -- considering these Poisson structures through their induced divisor -- allows us to  study them using Lie algebroids which are tailor-made to their degeneracy. In this paper we develop a framework for the study of Poisson structures of divisor-type. Below we outline the ideas underlying this framework, and mention our main results.\n\nThe techniques and language developed in this paper are used to various establish results for almost-regular Poisson structures, including the computation of Poisson cohomology \\cite{KlaasseLanius18}, determining the adjoint symplectic groupoids integrating them \\cite{KlaasseLi18}, and developing topological obstructions to their existence \\cite{Klaasse18three}. Moreover, this framework was used in \\cite{CavalcantiKlaasse18} in the development of Gompf--Thurston methods for symplectic Lie algebroids.\n\\subsection*{Poisson structures of divisor-type}\nIn this paper we adapt the language of divisors to the context of smooth manifolds, following their use in \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}. A divisor consist of a line bundle together with a section whose zero set is nowhere dense. Each such pair $(L,s)$ specifies a divisor ideal $I_s := s(\\Gamma(L^*)) \\subseteq C^\\infty(X)$. Abstractly, divisor ideals are locally principal ideals $I$ for which the support $Z(I) \\subseteq X$ of the quotient sheaf $C^\\infty(X) \/ \\mathcal{I}$ is nowhere dense, where\n\\begin{equation*}\n\tZ(I) = \\{x \\in X \\, | \\, f(x) = 0 \\text{ for all }f \\in \\mathcal{I}_x\\}.\n\\end{equation*}\nThese ideals keep track of degenerative behavior along this support, and are an extension of the concept of a Cartier divisor in algebraic and complex geometry to the smooth setting.\n\nPoisson structures naturally lead to divisors: Given $\\pi \\in {\\rm Poiss}(X)$ such that $\\dim(X) = 2n$, consider its Pfaffian $\\wedge^n \\pi \\in \\Gamma(\\det(TX))$. Note that $\\pi$ is nondegenerate exactly at those points $x \\in X$ where $\\wedge^n \\pi_x \\neq 0$. Thus $\\pi$ is generically nondegenerate if and only if the pair $(\\det(TX),\\wedge^n \\pi)$ is a divisor. In this case we denote this pair by ${\\rm div}(\\pi)$, with associated divisor ideal $I_\\pi \\subseteq C^\\infty(X)$. This ideal captures the behavior of $\\pi$ along $Z(\\pi) = Z(I_\\pi)$, called the degeneracy locus of $\\pi$, consisting of points where the rank of $\\pi$ is not maximal. \n\nIn general, given $\\pi \\in {\\rm Poiss}(X)$, the subset $X_{\\pi,{\\rm reg}} \\subseteq X$ of points where $\\pi$ is regular is open and dense. In contrast, the subset $X_{\\pi, {\\rm max}} \\subseteq X_{\\pi,{\\rm reg}}$ of points of maximal rank is not. Given an integer $m \\geq 0$, a Poisson structure is of $m$-divisor-type if it is generically of rank $2m$, and its failure to be regular is governed by a divisor. More precisely, for its partial Pfaffians we demand that $\\wedge^{m+1} \\pi \\equiv 0$ and there exists a line subbundle $K \\subseteq \\wedge^{2m} TX$ such that $(K, \\wedge^m \\pi)$ is a divisor, which provides a divisor ideal $I_\\pi$. A consequence of this definition is that $X_{\\pi,{\\rm max}} = X_{\\pi,{\\rm reg}}$ is open and dense, so that $\\pi$ is generically regular of rank $2m$. We stress, however, that not every generically regular Poisson structure is of $m$-divisor-type. This definition is closely related to almost-regularity as defined in \\cite{AndroulidakisZambon17}, as we discuss in \\autoref{sec:almostregularpoisson}.\n\nOne class of generically-nondegenerate Poisson structures is that of log-Poisson structures \\cite{GuilleminMirandaPires14,MarcutOsornoTorres14,Cavalcanti17,GualtieriLi14}, (also called log- or $b$-symplectic). Another class is given by elliptic Poisson structures \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18}, which arise (partially) as the Poisson structures underlying stable generalized complex structures. Further examples include $b^k$-Poisson structures \\cite{Scott16,GuilleminMirandaWeitsman17,Lanius16} and scattering Poisson structures \\cite{Lanius16}. Moreover, in this paper we introduce a novel class called elliptic-log Poisson structures (a more in-depth study of them is deferred to \\cite{KlaasseLanius18}). Each of these Poisson structures of divisor-type has a generically-regular analogue. In particular, there are $m$-log-Poisson structures of generic rank $2m$ (called log-f Poisson structures in \\cite{AndroulidakisZambon17}).\n\nBy keeping track of the divisor associated to a Poisson structure $\\pi$ of divisor-type, we can define almost-injective Lie algebroids $\\mathcal{A} \\to X$ whose degeneracy is similar or identical to that of $\\pi$. These Lie algebroids are candidates for a lifting procedure, whereby $\\pi$ is (partially) desingularized to a Lie algebroid Poisson structure. We first describe such Lie algebroids.\n\\subsection*{Lie algebroids of divisor-type}\nDivisor ideals can also be used to capture the degeneracy of Lie algebroids. This leads to Lie algebroids of divisor-type, which are almost-injective Lie algebroids for which the degeneration of the anchor map is controlled in a precise sense:\n\nGiven a Lie algebroid $\\mathcal{A} \\to X$, one can view its anchor as a section $\\rho_{A} \\in \\Gamma(\\mathcal{A}^* \\otimes TX)$. If ${\\rm rank}(\\mathcal{A}) = \\dim(X)$, then the zero locus of $\\det(\\rho_\\mathcal{A})$ consists of those points where $\\rho_{A}$ is not an isomorphism. Thus $\\mathcal{A}$ is almost-injective if and only if the pair $(\\det(\\mathcal{A}^*) \\otimes \\det(TX), \\det(\\rho_{A}))$ is a divisor, which we denote by ${\\rm div}(\\mathcal{A})$. In fact, divisors can be associated to any bundle map between bundles of the same rank, so that ${\\rm div}(\\mathcal{A}) = {\\rm div}(\\rho_{A})$. With this in mind, a Lie algebroid $\\mathcal{A} \\to X$ is of $m$-divisor-type for some $m \\geq 0$ if there exists a Lie subalgebroid $D \\subseteq TX$ of rank $m$ so that $\\rho_{A}(\\mathcal{A}) \\subseteq D$, and for which the induced Lie algebroid morphism $\\varphi_{\\mathcal{A}}\\colon \\mathcal{A} \\to D$ is of divisor-type. This forces that $\\mathcal{A}$ is of rank $m$, and that it is almost-injective.\n\nAbove we saw how a Lie algebroid $\\mathcal{A}$ has an associated divisor ${\\rm div}(\\mathcal{A})$, and consequently leads to a divisor ideal $I_\\mathcal{A} \\subseteq C^\\infty(X)$. There are also reverse procedures, where divisor ideals are used to construct Lie algebroids of divisor-type. The first of these proceeds as follows: Given a divisor ideal $I \\subseteq C^\\infty(X)$, consider the module of derivations preserving the ideal $I$\n\\begin{equation*}\n\t\\Gamma(TX)_I := \\{V \\in \\Gamma(TX) \\, | \\, \\mathcal{L}_V I \\subseteq I\\}.\n\\end{equation*}\nIf this module is projective, by the Serre--Swan theorem there is a unique almost-injective Lie algebroid $TX_I \\to X$ such that $\\Gamma(TX_I) \\cong \\Gamma(TX)_I$. This is called the ideal Lie algebroid associated to $I$, and we call $I$ projective if $\\Gamma(TX)_I$ is projective. The Lie algebroid $TX_I$ is of divisor-type, with divisor ideal $I_{TX_I}$. Here $I$ and $I_{TX_I}$ might differ, but are often related. This construction recovers the (complex) log- and elliptic tangent bundles \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18,GuilleminMirandaPires14,MarcutOsornoTorres14,Melrose93}.\n\nA second method to construct Lie algebroids  from divisors is the following: Let $\\mathcal{A} \\to X$ be a Lie algebroid, and consider a divisor ideal $I \\subseteq C^\\infty(X)$ whose support $Z(I)$ is a smooth submanifold. Given a Lie subalgebroid $\\mathcal{B}$ of $\\mathcal{A}$ which is supported on $Z(I)$, one can perform elementary modification to construct a new Lie algebroid $[\\mathcal{A}\\:(\\mathcal{B},I)]$ which is isomorphic to $\\mathcal{A}$ outside of $Z(I)$. In fact, there is an induced Lie algebroid morphism $\\varphi\\colon [\\mathcal{A}\\:(\\mathcal{B},I)] \\to \\mathcal{A}$ which is of divisor-type with divisor ideal $I_\\varphi = I^k$, where $k$ is the corank of $\\mathcal{B}$ with respect to $\\mathcal{A}$. This procedure extends the so-called modification procedures (or rescaling) performed in the case where $I = I_Z$, the vanishing ideal of a smooth hypersurface $Z \\subseteq X$ (see \\cite{Li13,GualtieriLi14,Lanius16,Melrose93}).\n\\subsection*{Lifting Poisson structures}\nPoisson structures and Lie algebroids of divisor-type are connected through the process of lifting. Given a Lie algebroid $\\mathcal{A} \\to X$, an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}} \\in {\\rm Poiss}(\\mathcal{A})$ is a section $\\pi_{A} \\in \\Gamma(\\wedge^2 \\mathcal{A})$ satisfying $[\\pi_{\\mathcal{A}},\\pi_{\\mathcal{A}}]_{\\mathcal{A}} = 0$, where $[\\cdot,\\cdot]_\\mathcal{A}$ is the Schouten bracket on $\\Gamma(\\wedge^\\bullet \\mathcal{A})$ extending the Lie bracket on $\\Gamma(\\mathcal{A})$. Such a structure turns the tuple $(\\mathcal{A},\\mathcal{A}^*,\\pi_{\\mathcal{A}})$ into a triangular Lie bialgebroid (\\cite{KosmannSchwarzbach95,KosmannSchwarzbachLaurentGengoux05,LiuWeinsteinXu97,MackenzieXu94}). Given $\\pi \\in {\\rm Poiss}(X)$, we say that $\\pi$ is of $\\mathcal{A}$-type if it admits an $\\mathcal{A}$-lift, i.e.\\ an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}}$ such that $\\rho_{\\mathcal{A}}(\\pi_{\\mathcal{A}}) = \\pi$.\n\nIf both $\\mathcal{A}$ and $\\pi$ are of divisor-type, then so is $\\pi_{\\mathcal{A}}$, and their divisor ideals are related:\n\\begin{equation*}\n\tI_\\pi = I_{\\pi_{\\mathcal{A}}} \\cdot I_\\mathcal{A}.\n\\end{equation*}\nIn other words, part of the degeneracy of $\\pi$ is absorbed into that of the anchor of $\\mathcal{A}$, making its $\\mathcal{A}$-lift $\\pi_{\\mathcal{A}}$ necessarily more regular. Because of the nature of the Lie algebroids considered in this paper, it is possible to use symplectic techniques to study nondegenerate $\\mathcal{A}$-Poisson structures. Indeed, a nondegenerate $\\mathcal{A}$-Poisson structure is dual to an $\\mathcal{A}$-symplectic structure, making $\\mathcal{A}$ into a symplectic Lie algebroid \\cite{NestTsygan01}. For example, Moser methods are available for such symplectic Lie algebroids $\\mathcal{A}$ of divisor-type (see \\cite[Theorem 5.5]{KlaasseLanius18}), which lead to Darboux-type normal form results for $\\pi_{\\mathcal{A}}$, and hence for the underlying $\\pi \\in {\\rm Poiss}(X)$.\n\nGiven a Poisson structure $\\pi$ of divisor-type, we would like to find a Lie algebroid $\\mathcal{A}$ for which $I_\\mathcal{A} = I_\\pi$, so that any $\\mathcal{A}$-lift of $\\pi$ will be nondegenerate. Sometimes such Lie algebroids can be constructed from the divisor ideal $I_\\pi$. This lifting process necessitates the study of Poisson and symplectic structures in a general Lie algebroid $\\mathcal{A}$, which is the aim of this paper.\n\nWhile most of this paper is written in terms of $\\mathcal{A}$-Lie algebroids and $\\mathcal{A}$-Poisson structures, our main interest is to study Poisson structures $\\pi \\in {\\rm Poiss}(X)$ in the standard sense. The $\\mathcal{A}$-notions have to be developed for their own sake, but are mainly used as auxiliary tools through the process of lifting. Note that there can be multiple non-isomorphic Lie algebroids to which one can nondegenerately lift a given Poisson structure (see \\cite{KlaasseLanius18}).\n\\subsection*{Results}\nThe main aim of this paper is to develop an effective framework for the study of Poisson structures of divisor-type. Because of this we introduce and study:\n\\begin{itemize}\n\t\\item Divisor ideals keeping track of degeneracies;\n\t\\item Lie algebroids of divisor-type, with emphasis on:\n\t%\n\t\\begin{itemize}\n\t\t\\item Ideal Lie algebroids of derivations preserving a divisor ideal;\n\t\t\\item Elementary modification of Lie algebroids using divisor ideals;\n\t\\end{itemize}\n\t\\item Poisson structures of divisor-type, leading to:\n\t%\n\t\\begin{itemize}\n\t\t\\item The process of lifting Poisson structures to their $\\mathcal{A}$-Poisson counterparts;\n\t\\end{itemize}\n\t\\item Residue maps to extract information along the degeneraci loci;\n\t\\item The relation between the lifting Lie algebroid and the modular foliation.\n\\end{itemize}\nHere we highlight several aspects of this undertaking that we provide in this paper. \n\nThe first result concerns lifting for Poisson structures of divisor-type to the ideal Lie algebroids they define. This recaptures the lifting results for log- and elliptic Poisson structures of \\cite{GuilleminMirandaPires14} and \\cite{CavalcantiGualtieri18}. The following is \\autoref{thm:ipoissonaitype}, on lifting to the ideal Lie algebroid $TX_I$.\n\\begin{thmalpha}[Lifting]\\label{thm:introlifting} Let $\\pi \\in {\\rm Poiss}(X)$ be of projective $I$-divisor-type with $TX_I^*$ admitting a local basis of closed sections. Then $\\pi$ is of $TX_I$-type, and of nondegenerate $TX_I$-type if and only if $I$ is standard, i.e.\\ the divisor ideal of $TX_I$ satisfies $I_{TX_I} = I$.\n\\end{thmalpha}\nThe next theorem discusses the residue maps of Lie algebroids $\\mathcal{A} \\to X$. Given an invariant submanifold $D \\subseteq X$ on which the module $(\\rho_\\mathcal{A}|_D)(\\Gamma(\\mathcal{A}|_D)) \\cong \\Gamma(\\mathcal{B})$ is projective, there is an induced surjective morphism $\\widetilde{\\rho}_\\mathcal{A}|_D\\colon \\mathcal{A}|_D \\to \\mathcal{B}$ resulting in a Lie algebroid extension sequence\n\\begin{equation*}\n\t0 \\to E \\to \\mathcal{A}|_D \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nThe resulting residue maps are used in computing the Lie algebroid cohomology of $\\mathcal{A}$, and to extract information from $\\mathcal{A}$-symplectic structures. In \\autoref{sec:residues} we conceptually describe the residue maps found in  \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18,GuilleminMirandaPires14,MarcutOsornoTorres14,Scott16}, and are used in \\cite{KlaasseLanius18}. The following is \\autoref{thm:residues}.\n\\begin{thmalpha}[Residue maps]\\label{thm:introresidues} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $D \\subseteq X$ be a projective $\\mathcal{A}$-invariant submanifold with induced Lie algebroid $\\mathcal{B} \\to D$. Then there exists a residue map\n\\begin{equation*}\n\t{\\rm Res}_D\\colon \\Omega^\\bullet(X,\\mathcal{A}) \\to \\Omega^{\\bullet-\\ell}(D,\\mathcal{B}; \\det(E^*)), \\qquad \\text{where } \\ell = {\\rm rank}(E).\n\\end{equation*}\nThis is a cochain morphism for the natural $\\mathcal{B}$-representation on $\\det(E^*)$, hence descends to\n\\begin{equation*}\n\t[{\\rm Res}_D]\\colon H^\\bullet(X,\\mathcal{A}) \\to H^{\\bullet-\\ell}(D,\\mathcal{B}; \\det(E^*)).\n\\end{equation*}\n\\end{thmalpha}\nWe further discuss the relation between the lifting process and the modular foliation \\cite{GualtieriPym13}, which arises as the sheaf of submodules of $\\Gamma(TX)$ spanned by the modular and Hamiltonian vector fields. The symplectic foliation $\\mathcal{F}_\\pi$ and the modular foliation $\\mathcal{F}_{\\pi,{\\rm mod}}$ of a Poisson structure $\\pi \\in {\\rm Poiss}(X)$ interact with any lifting Lie algebroid. The following is \\autoref{thm:modfoliation}.\n\\begin{thmalpha}[Modular foliation]\\label{thm:intromodfoliation} Let $\\pi \\in {\\rm Poiss}(X)$ be of $\\mathcal{A}$-type for a Lie algebroid $\\mathcal{A} \\to X$. Then $\\mathcal{F}_{\\pi} \\subseteq \\mathcal{F}_{\\pi, {\\rm mod}} \\subseteq \\rho_\\mathcal{A}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)$, and the orbits of $\\mathcal{A}$ are $\\pi$-Poisson submanifolds.\n\\end{thmalpha}\nThis result is most interesting when $\\mathcal{A}$ is almost-injective, i.e.\\ when $\\rho_\\mathcal{A}$ is injective on sections. In this case \\autoref{thm:intromodfoliation} gives an intuitive picture of the lifting procedure. It also helps in determining prospective candidate Lie algebroids to lift Poisson structures to.\n\\begin{rem} In this paper we do not discuss Dirac geometry nor generalized complex geometry. Their interaction with divisors and the lifting procedure are instead explored in \\cite{Klaasse18}, see also \\cite{CavalcantiKlaasse18,CavalcantiGualtieri18, Klaasse17}. We do not discuss the Lie groupoids associated to the Lie algebroids we consider, which are all integrable as they are almost-injective \\cite{CrainicFernandes03,CrainicFernandes04,Debord01}. See \\cite{GualtieriLi14,KlaasseLi18,Li13} for explicit constructions of integrations in the setting of log- and elliptic Poisson manifolds.\n\\end{rem}\n\\subsection*{Acknowledgements} This work is partially based on the author's Ph.D.\\ thesis \\cite{Klaasse17}, and was supported by ERC grant 646649, and by VIDI grant 639.032.221 from NWO, the Netherlands Organisation for Scientific Research. The author would like to thank Melinda Lanius, Rui Loja Fernandes, Ioan M{\\u{a}}rcu{\\c{t}}, Brent Pym, Marco Zambon, and especially his Ph.D.\\ advisor Gil Cavalcanti for useful discussions. Moreover, the author would like to thank the University of Illinois at Urbana--Champaign for its hospitality in early 2017 during a part of this project.\n\\subsection*{Organization of the paper}\nThis paper is structured as follows. In \\autoref{sec:divonmanifolds} we discuss the theory of divisors on smooth manifolds, emphasizing especially the notion of a divisor ideal. In \\autoref{sec:liealgebroidsdivtype} we then turn to Lie algebroids of divisor-type, and how divisor ideals can be used to construct them. Then, in \\autoref{sec:poissondivtype} we consider Poisson structures of divisor-type in general Lie algebroids, and discuss the process of lifting Poisson structures along Lie algebroid morphisms. In \\autoref{sec:residues} we then discuss residue maps for Lie algebroids and Poisson modules, and how these can be used to study symplectic Lie algebroids. We further study the modular foliation of a Poisson manifold, and how it interacts with the lifting procedure.\n\n\\section{Lie algebroids of divisor-type}\n\\label{sec:liealgebroidsdivtype}\nIn this section we discuss Lie algebroids and their interaction with divisors, starting in \\autoref{sec:liealgebroids} with how Lie algebroids can specify divisors through the degeneracy of their anchor. \\autoref{sec:aliealgebroids} through \\autoref{sec:singularliealgebroids} discuss several further concepts that we will use.\n\nWe describe two processes by which Lie algebroids can be constructed using divisors. The first of these gives rise to ideal Lie algebroids (in \\autoref{sec:idealliealgebroids}), which are those Lie algebroids whose sections are those preserving a given ideal of functions. The second is that of elementary modification (in \\autoref{sec:elementarymodifications}). Here one uses a injective morphism from, or a surjective comorphism to, a subbundle supported on the zero locus of a smooth divisor ideal. \n\\subsection{Lie algebroids}\n\\label{sec:liealgebroids}\nFor general information regarding Lie algebroids, see \\cite{Mackenzie05}. A vector bundle $\\mathcal{A} \\to X$ is \\emph{anchored} if it has a bundle map $\\rho_{A}\\colon \\mathcal{A} \\to TX$ called its \\emph{anchor}. We refer to morphisms of anchored bundles (those intertwining the anchor maps) by \\emph{anchored morphisms}.\n\\begin{defn} \\label{defn:liealgebroid} A \\emph{Lie algebroid} $(\\mathcal{A},[\\cdot,\\cdot]_{\\mathcal{A}},\\rho_\\mathcal{A})$ is an anchored bundle with a Lie bracket $[\\cdot,\\cdot]_{\\mathcal{A}}$ on $\\Gamma(\\mathcal{A})$ such that $[v,f w]_{\\mathcal{A}} = f [v,w]_{\\mathcal{A}} + \\mathcal{L}_{\\rho_{\\mathcal{A}}(v)} f \\cdot w$ for all $v, w \\in \\Gamma(\\mathcal{A})$, $f \\in C^\\infty(X)$.\n\\end{defn}\nThe bracket $[\\cdot,\\cdot]_\\mathcal{A}$ of a Lie algebroid $\\mathcal{A}$ provides the algebra $\\Omega^\\bullet(\\mathcal{A}) = \\Gamma(\\wedge^\\bullet \\mathcal{A}^*)$ with a differential $d_\\mathcal{A}$ using the Koszul formula. This allows for a simple definition of morphisms.\n\\begin{defn} A \\emph{Lie algebroid morphism} is a bundle morphism $(\\varphi,f)\\colon \\mathcal{A} \\to \\mathcal{A}'$ for which the induced map $\\varphi^*\\colon \\Omega^*(\\mathcal{A}') \\to \\Omega^*(\\mathcal{A})$ is a chain map, i.e.\\ satisfies $\\varphi^* \\circ d_{\\mathcal{A}'} = d_\\mathcal{A} \\circ \\varphi^*$.\n\\end{defn}\n\\begin{rem} Given $v \\in \\Gamma(\\mathcal{A})$, its \\emph{$\\mathcal{A}$-Lie derivative} is the operator $\\mathcal{L}_v = \\{\\iota_v, d_\\mathcal{A}\\}$ on $\\Omega^\\bullet(\\mathcal{A})$, where $\\{\\cdot,\\cdot\\}$ is the graded commutator. It satisfies the usual Cartan relations (see e.g.\\ \\cite{Marle08}).\n\\end{rem}\nLie algebroid morphisms are in particular anchored morphisms, i.e.\\ they satisfy the relation $\\rho_{\\mathcal{A}'} \\circ \\varphi = Tf \\circ \\rho_\\mathcal{A}$. If $(\\varphi,f)\\colon \\mathcal{A} \\to \\mathcal{A}'$ is a base-preserving anchored morphism, it is a Lie algebroid morphism if and only if it moreover induces a Lie algebra homomorphism $\\varphi\\colon \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A}')$. Note that the anchor map $\\rho_\\mathcal{A}$ is a Lie algebroid morphism from $\\mathcal{A}$ to $TX$. For base-preserving Lie algebroid morphisms, we keep track of where it is an isomorphism.\n\\begin{defn} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ be a Lie algebroid morphism. Its \\emph{isomorphism locus} is the open subset $X_\\varphi \\subseteq X$ where $\\varphi$ is an isomorphism.\n\\end{defn}\nThe complement of the isomorphism locus is denoted by $Z_\\varphi := X \\backslash X_\\varphi$, and is called the \\emph{degeneracy locus} of $\\varphi$. For a Lie algebroid $\\mathcal{A}$ we define $X_{\\mathcal{A}} := X_{\\rho_\\mathcal{A}}$ and refer to this as the isomorphism locus of $\\mathcal{A}$, and similarly for $Z_\\mathcal{A}$. Note that whenever $X_\\varphi$ is nonempty, we have ${\\rm rank}(\\mathcal{A}) = {\\rm rank}(\\mathcal{A}')$ because $X_\\varphi$ is open. Between bundles of the same rank, the condition that $X_\\varphi$ is dense is equivalent to injectivity of the induced map on sections $\\varphi\\colon \\Gamma(\\mathcal{A}') \\to \\Gamma(\\mathcal{A})$.\n\\begin{rem} Given two composable base-preserving Lie algebroid morphisms $\\varphi$ and $\\varphi'$, we have $X_{\\varphi \\circ \\varphi'} \\supseteq X_\\varphi \\cap X_{\\varphi'}$. If we further assume that all Lie algebroids involved are of the same rank (e.g.\\ by each isomorphism locus being nonempty), this becomes an equality. In particular, in this case, if the isomorphism locus $X_{\\varphi \\circ \\varphi'}$ is dense, then so are $X_\\varphi$ and $X_{\\varphi'}$.\n\\end{rem}\nGiven a Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$, we can take determinants to obtain $\\det(\\varphi)\\colon \\det(\\mathcal{A}') \\to \\det(\\mathcal{A})$. This specifies a section $\\det(\\varphi) \\in \\Gamma(\\det(\\mathcal{A}'^*) \\otimes \\det(\\mathcal{A}))$. It is now readily verified that $\\varphi$ has dense isomorphism locus if and only if this defines a divisor.\n\\begin{defn}\\label{defn:morphismdivisortype} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ be a Lie algebroid morphism with dense isomorphism locus. The \\emph{divisor associated to $\\varphi$} is ${\\rm div}(\\varphi) = (\\det(\\mathcal{A}'^*) \\otimes \\det(\\mathcal{A}), \\det(\\varphi))$, with the associated divisor ideal being denoted by $I_\\varphi$. Given a Lie algebroid $\\rho_\\mathcal{A}\\colon \\mathcal{A} \\to TX$ with dense isomorphism locus, we define ${\\rm div}(\\mathcal{A}) := {\\rm div}(\\rho_\\mathcal{A})$ and $I_\\mathcal{A} := I_{\\rho_\\mathcal{A}}$, and call $\\mathcal{A}$ of \\emph{divisor-type}.\n\\end{defn}\nGiven a divisor ideal $I$ we call a base-preserving Lie algebroid morphism $\\varphi$ \\emph{of $I$-divisor-type} if $I_\\varphi = I$, and similarly for Lie algebroids $\\mathcal{A}$. It is immediate that composition of morphisms of divisor-type corresponds to taking products of their divisors (hence their divisors ideals): after noting that $\\det(\\varphi' \\circ \\varphi) = \\det(\\varphi') \\circ \\det(\\varphi)$ we obtain the equality $I_{\\varphi' \\circ \\varphi} = I_{\\varphi'} \\cdot I_\\varphi$.\n\\begin{prop}\\label{prop:divisorcomposition} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and $(\\varphi',{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}''$ be Lie algebroid morphisms of divisor-type. Then $\\varphi' \\circ \\varphi$ is of divisor-type, and ${\\rm div}(\\varphi' \\circ \\varphi) \\cong {\\rm div}(\\varphi') \\otimes {\\rm div}(\\varphi)$.\n\\end{prop}\nIn the generically-regular setting, we define a Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ to be of \\emph{$m$-divisor-type} for some $m \\geq 0$ if there exists a regular Lie subalgebroid $D_\\mathcal{A} \\subseteq \\mathcal{A}$ of rank $m$ (see \\autoref{sec:liesubalgds}, in particular \\autoref{exa:adistribution}) such that $\\varphi$ factors through the inclusion $\\varphi_{D_\\mathcal{A}}\\colon D_{\\mathcal{A}} \\to \\mathcal{A}$, i.e.\\ $\\varphi = \\varphi_{D_\\mathcal{A}} \\circ \\varphi_{\\mathcal{A}'}$ for some Lie algebroid morphism $\\varphi_{\\mathcal{A}'}\\colon \\mathcal{A}' \\to D_{\\mathcal{A}}$, and $\\varphi_{\\mathcal{A}'}$ is of divisor-type, with divisor given by ${\\rm div}(\\varphi_{\\mathcal{A}'}) = (\\det(\\mathcal{A}'^*) \\otimes \\det(D_{\\mathcal{A}}), \\det(\\varphi_{\\mathcal{A}'}))$. Alternatively, we can demand the existence of a line subbundle $K_\\mathcal{A} \\subseteq \\wedge^m \\mathcal{A}$ such that $\\wedge^m \\varphi\\colon \\det(\\mathcal{A}') \\to K_\\mathcal{A}$ and $(\\det(\\mathcal{A}'^*)\\otimes K_\\mathcal{A}, \\wedge^m \\varphi)$ is a divisor. These formulations are related via $\\det(D_\\mathcal{A}) = K_\\mathcal{A}$. We then define $\\mathcal{A}$ to be of $m$-divisor-type if its anchor $\\rho_{A}\\colon \\mathcal{A} \\to TX$ is.\n\nWe introduce several further adjectives to describe the behavior of the anchor.\n\\begin{defn}\\label{defn:lalgebroidadjective} Let $\\mathcal{A} \\to X$ be a Lie algebroid. Then $\\mathcal{A}$ is \\emph{regular} if $\\rho_\\mathcal{A}$ has constant rank, \\emph{transitive} if $\\rho_\\mathcal{A}$ is surjective, and \\emph{totally intransitive} if $\\rho_\\mathcal{A} \\equiv 0$. Further, $\\mathcal{A}$ is \\emph{almost-injective} \\cite{Debord01} if $\\rho_\\mathcal{A}\\colon \\Gamma(\\mathcal{A}) \\to \\Gamma(TX)$ is injective, and \\emph{projective} if $\\rho_{A}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)$ is projective.\n\\end{defn}\nThese properties are related: transitivity implies regularity, which in turn implies projectivity. If ${\\rm rank}(\\mathcal{A}) = \\dim X$, then $\\mathcal{A}$ is almost-injective if and only if it is of divisor-type.\n\\begin{defn}\\label{defn:laisotropy} The \\emph{isotropy} of $\\mathcal{A} \\to X$ at $x \\in X$ is given by the subspace $\\ker \\rho_{\\mathcal{A},x} \\subseteq \\mathcal{A}_x$.\n\\end{defn}\nThis subspace makes sense for any anchored vector bundle, but for Lie algebroids the isotropy is a Lie algebra, inheriting a bracket from $\\mathcal{A}$. The isotropy Lie algebras $\\ker \\rho_{\\mathcal{A},x}$ may vary in dimension as $x \\in X$ varies, so that in general these may not form a subbundle of $\\mathcal{A}$. A Lie algebroid (or anchored) morphism $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ restricts to a map $\\varphi\\colon \\ker \\rho_{\\mathcal{A},x} \\to \\ker \\rho_{\\mathcal{A}',f(x)}$ between pointwise isotropies for all $x \\in X$.\n\nThe image of any Lie algebroid under the anchor map determines a singular foliation\n\\begin{equation*}\n\t\\mathcal{F}_\\mathcal{A} := \\rho_{\\mathcal{A}}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)\n\\end{equation*}\nin the sense of Androulidakis--Skandalis \\cite{AndroulidakisSkandalis09}. Its leaves, i.e.\\ the associated maximal immersed submanifolds $\\mathcal{O}$ satisfying $T_x \\mathcal{O} = {\\rm im}\\, \\rho_{\\mathcal{A},x}$ for all $x \\in \\mathcal{O}$, are called the \\emph{orbits} of $\\mathcal{A}$.\n\nFor regular Lie algebroids (hence for transitive ones), the kernel of $\\rho_\\mathcal{A}$ is of constant dimension, so that it is a subbundle of $\\mathcal{A}$. If $\\mathcal{A}$ is transitive, the isotropy Lie algebras together exhibit $\\mathcal{A}$ as an abelian extension of $TX$. In other words, there is a short exact sequence:\n\\begin{equation*}\n\t0 \\to \\ker \\rho_\\mathcal{A} \\to \\mathcal{A} \\to TX \\to 0.\n\\end{equation*}\nA similar sequence exists for projective Lie algebroids. In this case (as in \\autoref{sec:singularliealgebroids}), there exists an almost-injective Lie algebroid $\\mathcal{B} \\to X$ with $\\Gamma(\\mathcal{B}) \\cong \\rho_{A}(\\Gamma(\\mathcal{A}))$ which is unique up to isomorphism. This comes with an induced Lie algebroid morphism $\\widetilde{\\rho}_\\mathcal{A}\\colon \\mathcal{A} \\to \\mathcal{B}$ induced by the map on sections $\\rho_{A}\\colon \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{B})$. This results in a short exact sequence of Lie algebroids:\n\\begin{equation*}\n\t0 \\to \\ker \\widetilde{\\rho}_\\mathcal{A} \\to \\mathcal{A} \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nUnlike the transitive case, the Lie algebroid $\\ker \\widetilde{\\rho}_\\mathcal{A}$ need not be abelian. Following \\cite[Definition 1.10]{AndroulidakisZambon17}, we make the following definition in this case.\n\\begin{defn}\\label{defn:germinalisotropy} The \\emph{germinal isotropy} of $\\mathcal{A} \\to X$ at $x \\in X$ is the subspace $\\ker \\widetilde{\\rho}_{\\mathcal{A},x} \\subseteq \\mathcal{A}_x$.\n\\end{defn}\nIn general one has $\\ker \\widetilde{\\rho}_{\\mathcal{A},x} \\subseteq \\ker \\rho_{\\mathcal{A},x}$. This definition still makes sense when the module $\\mathcal{F}_\\mathcal{A} \\subseteq \\Gamma(TX)$ is projective: if it is not, we can still consider the vector spaces $\\mathcal{F}_\\mathcal{A} \/ I_x \\cdot \\mathcal{F}_\\mathcal{A}$. We will return to the germinal isotropy in \\autoref{sec:singularliealgebroids}.\\\\\n\nLet $\\mathcal{A} \\to X$ be a Lie algebroid with dense isomorphism locus $X_\\mathcal{A}$. Then the inclusion $i\\colon X_{\\mathcal{A}} \\hookrightarrow X$ gives a bijection $\\rho_{\\mathcal{A}}^*\\colon \\Omega^k(X_{\\mathcal{A}}) \\to \\Omega^k_{\\mathcal{A}}(X_{\\mathcal{A}})$ for all $k$. Consequently, we can view $\\mathcal{A}$-forms as smooth forms with certain ``singular'' behavior at its degeneracy locus $Z_\\mathcal{A}$. The isomorphism given by the restricted anchor map $\\rho_\\mathcal{A}\\colon \\mathcal{A}|_{X_\\mathcal{A}} \\to TX|_{X_\\mathcal{A}}$ also implies the following.\n\\begin{prop}\\label{prop:denseisolocusstrongmap} Let $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ be an anchored morphism between Lie algebroids of divisor-type such that $f^{-1}(X_{\\mathcal{A}'}) = X_\\mathcal{A}$. Then $\\varphi \\equiv Tf$ on sections, and $\\varphi$ is a Lie algebroid morphism.\n\\end{prop}\n\\bp When restricted to the isomorphism loci, we have that $\\varphi \\equiv Tf$ as bundle maps after intertwining with the anchors. Because the isomorphism loci are dense, this means that $\\varphi \\equiv Tf$ on sections, which holds globally on $X$. Because $\\varphi$ is a bundle morphism, the map $\\varphi^*$ is an algebra morphism. In the isomorphism loci, $\\varphi$ must equal $Tf$, and $(Tf,f)$ is a Lie algebroid morphism between $TX$ and $TX'$, which means that $f^*$ is a chain map. By the density of $X_{\\mathcal{A}}$ in $X$, we conclude that the map $\\varphi^*$ is a chain map everywhere.\n\\end{proof}\nThus, for Lie algebroids as \\autoref{prop:denseisolocusstrongmap}, to determine whether there is a Lie algebroid morphism $(\\varphi, f)$, one can check whether $f^*$ extends to a map $\\varphi^*$ on forms. This in turn follows if it holds on generators, so it suffices to verify that $f^*$ extends to a map $\\varphi^*\\colon \\Omega^1(\\mathcal{A}') \\to \\Omega^1(\\mathcal{A})$.\n\\subsection{\\texorpdfstring{$\\mathcal{A}$}{A}-Lie algebroids}\n\\label{sec:aliealgebroids}\nWe now introduce what we call $\\mathcal{A}$-versions of Lie algebroids, where $\\mathcal{A} \\to X$ is a fixed Lie algebroid. The main idea is that these objects come equipped with a preferred Lie algebroid morphism onto $\\mathcal{A}$, called its $\\mathcal{A}$-anchor, through which its own anchor factors. Our interest later will be in those examples for which this $\\mathcal{A}$-anchor is of divisor-type.\n\\begin{defn}\\label{defn:aavala} An \\emph{$\\mathcal{A}$-anchored vector bundle} is an anchored vector bundle $E_\\mathcal{A} \\to X$ with an anchored morphism $\\varphi_{E_\\mathcal{A}} \\colon E_\\mathcal{A} \\to \\mathcal{A}$ called the \\emph{$\\mathcal{A}$-anchor} satisfying $\\rho_{E_\\mathcal{A}} = \\rho_\\mathcal{A} \\circ \\varphi_{E_\\mathcal{A}}$. In turn, an \\emph{$\\mathcal{A}$-Lie algebroid} is an $\\mathcal{A}$-anchored vector bundle $E_\\mathcal{A}$ equipped with a compatible Lie algebroid structure for which $\\varphi_{E_\\mathcal{A}}$ is a Lie algebroid morphism.\n\\end{defn}\nThis concept is useful especially when dealing with Lie algebroids whose sections form a submodule of $\\Gamma(\\mathcal{A})$, i.e.\\ those whose $\\mathcal{A}$-anchor $\\varphi_{E_\\mathcal{A}}\\colon \\Gamma(E_{\\mathcal{A}}) \\to \\Gamma(\\mathcal{A})$ is injective on sections. \\autoref{defn:morphismdivisortype} provides any $\\mathcal{A}$-Lie algebroid $\\mathcal{A}'$ with the notion of being of \\emph{$\\mathcal{A}$-divisor-type} by looking at the pair $(\\det(\\mathcal{A}'^*)\\otimes \\det(\\mathcal{A}), \\det(\\varphi_{\\mathcal{A}}))$, with its \\emph{$\\mathcal{A}$-divisor ideal} being $I_{\\varphi_{\\mathcal{A}}}$. Note also the relevance of \\autoref{prop:divisorcomposition} which shows that $I_{\\mathcal{A}'} = I_{\\varphi_{\\mathcal{A}}} \\cdot I_\\mathcal{A}$ when this makes sense.\n\\begin{rem} Any Lie algebroid $\\mathcal{A} \\to X$ is naturally both an $\\mathcal{A}$- and a $TX$-Lie algebroid.\n\\end{rem}\nRecall that a \\emph{comorphism of anchored vector bundles} $(\\varphi;f)\\colon (E,X) \\dashedrightarrow (E',X')$ consists of a base map $f\\colon X \\to X'$ and a vector bundle morphism $\\varphi\\colon f^* E' \\to E$ such that\n\\begin{equation*}\n\t\\rho_E (\\varphi^*(\\sigma)) \\sim_f \\rho_{E'}(\\sigma) \\qquad \\text{for all } \\sigma \\in \\Gamma(E').\n\\end{equation*}\n We will call these \\emph{anchored comorphisms} for short. An anchored comorphism gives a pullback map $\\varphi^*\\colon \\Gamma(E') \\to \\Gamma(E)$ on sections. A \\emph{Lie algebroid comorphism} $(\\varphi;f)\\colon (E,X) \\dashedrightarrow (E',X')$ is an anchored comorphism for which the induced map $\\varphi^*\\colon \\Gamma(E') \\to \\Gamma(E)$ preserves brackets, i.e.\\ satisfies $\\varphi^*[\\sigma,\\tau]_{E'} = [\\varphi^*(\\sigma),\\varphi^*(\\tau)]_{E}$ for all $\\sigma,\\tau \\in \\Gamma(E')$.\n\\begin{rem} Using the fact that Lie algebroid structures on $\\mathcal{A}$ are in one-to-one correspondence with linear Poisson structures on $\\mathcal{A}^*$, a Lie algebroid comorphism is alternatively defined as a bundle morphism between Lie algebroids whose dual morphism is a Poisson map.\n\\end{rem}\nThe $\\mathcal{A}$-version of the notions of morphism and comorphism are rather straightforward.\n\\begin{defn} An \\emph{$\\mathcal{A}$-anchored morphism} $(\\varphi',\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ consists of an anchored morphism $(\\varphi',f)\\colon (E_\\mathcal{A},X) \\to (E_{\\mathcal{A}'},X')$ and a compatible Lie algebroid morphism $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$, i.e.\\ such that we have the intertwining relation $\\varphi \\circ \\varphi_{E_\\mathcal{A}} = \\varphi_{E_\\mathcal{A}'} \\circ \\varphi'$.\n\t\nAn \\emph{$\\mathcal{A}$-Lie algebroid morphism} $(\\varphi',\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ is an $\\mathcal{A}$-anchored morphism for which $(\\varphi,f)\\colon (E_\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, X')$ is a Lie algebroid morphism.\n\\end{defn}\n\\begin{defn} An \\emph{$\\mathcal{A}$-anchored comorphism} $(\\varphi';\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\dashedrightarrow (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ consists of an anchored comorphism $(\\varphi';f)\\colon (E_\\mathcal{A},X) \\dashedrightarrow (E_{\\mathcal{A}'},X')$ and a compatible Lie algebroid morphism $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$, i.e.\\ such that $\\varphi_{E_\\mathcal{A}}(\\varphi^*(\\sigma)) \\sim_{\\varphi'} \\varphi_{E_\\mathcal{A}'}(\\sigma)$ for all $\\sigma \\in \\Gamma(E_{\\mathcal{A}'})$.  An \\emph{$\\mathcal{A}$-Lie algebroid comorphism} $(\\varphi';\\varphi,f)\\colon (E_\\mathcal{A},\\mathcal{A},X) \\dashedrightarrow (E_{\\mathcal{A}'}, \\mathcal{A}',X')$ is an $\\mathcal{A}$-anchored comorphism for which $(\\varphi;f)\\colon (E_\\mathcal{A},X) \\to (E_{\\mathcal{A}'}, X')$ is a Lie algebroid comorphism.\n\\end{defn}\nTo summarize, $\\mathcal{A}$-Lie algebroid (co)morphisms give rise to diagrams as follows:\n\\begin{center}\n\t\\begin{tikzpicture}\n\t\\begin{scope}[xshift=-200]\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t\\rho_{E_\\mathcal{A}}\\colon &[-3.1em] E_\\mathcal{A} & \\mathcal{A} & TX \\\\ \\rho_{E_{\\mathcal{A}'}}\\colon &[-3.2em] E_{\\mathcal{A}'} & \\mathcal{A}' & TX'\\\\};\n\t\\path[-stealth]\n\t(m-1-2) edge node [above] {$\\varphi_{E_\\mathcal{A}}$} (m-1-3)\n\t(m-1-2) edge node [left] {$\\varphi$} (m-2-2)\n\t(m-2-2) edge node [above] {$\\varphi_{E_{\\mathcal{A}'}}$} (m-2-3)\n\t(m-1-3) edge node [left] {$\\varphi'$} (m-2-3)\n\t(m-1-3) edge node [above] {$\\rho_\\mathcal{A}$} (m-1-4)\n\t(m-2-3) edge node [above] {$\\rho_{\\mathcal{A}'}$} (m-2-4)\n\t(m-1-4) edge node [left] {$T f$} (m-2-4);\n\t\\end{scope}\n\t%\n\t\\begin{scope}\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t\\rho_{E_\\mathcal{A}}\\colon &[-3.1em] E_\\mathcal{A} & \\mathcal{A} & TX\\\\ \\rho_{E_{\\mathcal{A}'}}\\colon &[-3.2em] E_{\\mathcal{A}'} & \\mathcal{A}' & TX'\\\\};\n\t\\path[-stealth]\n\t(m-1-2) edge node [above] {$\\varphi_{E_\\mathcal{A}}$} (m-1-3)\n\t(m-2-2) edge node [above] {$\\varphi_{E_{\\mathcal{A}'}}$} (m-2-3)\n\t(m-1-3) edge node [left] {$\\varphi'$} (m-2-3)\n\t(m-1-3) edge node [above] {$\\rho_\\mathcal{A}$} (m-1-4)\n\t(m-2-3) edge node [above] {$\\rho_{\\mathcal{A}'}$} (m-2-4)\n\t(m-1-4) edge node [left] {$T f$} (m-2-4);\n\t\\path[-stealth, dashed]\n\t(m-1-2) edge node [left] {$\\varphi$} (m-2-2);\n\t\\end{scope}\n\t\\end{tikzpicture}\n\\end{center}\n\\subsection{Fiber products}\n\\label{sec:algoperations}\nWe introduce the fiber product of Lie algebroids over the same base. This is a special case of the fiber product construction for Lie algebroids as found in \\cite{HigginsMackenzie90}.\n\\begin{defn}\\label{def:fiberproduct} Consider a Lie algebroid $\\mathcal{A} \\to X$ and $\\mathcal{A}$-Lie algebroids $(\\varphi_{\\mathcal{A}'}, {\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}$ and $(\\varphi_{\\mathcal{A}''}, {\\rm id}_X)\\colon \\mathcal{A}'' \\to \\mathcal{A}$. Assume that $\\varphi_{\\mathcal{A}'}$ and $\\varphi_{\\mathcal{A}''}$ are transverse in $\\mathcal{A}$, i.e.\\\n\\begin{equation*}\n\t\\varphi_{\\mathcal{A}'}(\\mathcal{A}') + \\varphi_{\\mathcal{A}''}(\\mathcal{A}'') = \\mathcal{A}\n\\end{equation*}\nat each point in $X$. The \\emph{fiber product} of $\\mathcal{A}'$ and $\\mathcal{A}''$ over $\\mathcal{A}$ is the Lie algebroid defined as\n\t%\n\t\\begin{equation*}\n\t\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A}'' = \\{(v,w) \\in \\mathcal{A}' \\times \\mathcal{A}'' \\, | \\varphi_{\\mathcal{A}'}(v) = \\varphi_{\\mathcal{A}''}(w) \\}.\n\t\\end{equation*}\n\t%\n\tThe anchor is inherited from either $\\mathcal{A}'$ or $\\mathcal{A}''$ and its bracket is generated by the expression $[v \\oplus v', w \\oplus w']_{\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A}''} = [v,w]_{\\mathcal{A}'} \\oplus [v',w']_{\\mathcal{A}''}$, for $v,w \\in \\Gamma(\\mathcal{A}')$ and $v',w' \\in \\Gamma(\\mathcal{A}'')$\n\\end{defn}\nThe projections onto $\\mathcal{A}'$ and $\\mathcal{A}''$ turn the fiber product $\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A}''$ into both an $\\mathcal{A}'$- and an $\\mathcal{A}''$-Lie algebroid. There is a natural isomorphism $\\mathcal{A}' \\times_{\\mathcal{A}} \\mathcal{A} \\cong \\mathcal{A}'$. If $\\mathcal{A}$ has dense isomorphism locus, recall that $Z_\\mathcal{A} = X \\backslash X_\\mathcal{A}$  is the degeneracy locus of $\\mathcal{A}$, and the support of its divisor.\n\\begin{prop}\\label{prop:fiberprodisolocus} Let $\\mathcal{A}, \\mathcal{A}' \\to X$ be Lie algebroids of divisor-type with disjoint supports. Then $\\rho_\\mathcal{A}$ and $\\rho_{\\mathcal{A}'}$ are transverse, and $\\mathcal{A} \\times_{TX} \\mathcal{A}'$ is of divisor-type with support $Z_\\mathcal{A} \\cup Z_{\\mathcal{A}'}$.\n\\end{prop}\n\\bp As $Z_\\mathcal{A}$ and $Z_{\\mathcal{A}'}$ are disjoint, pointwise either $\\rho_\\mathcal{A}$ or $\\rho_{\\mathcal{A}'}$ is an isomorphism, so that $\\rho_\\mathcal{A}$ and $\\rho_{\\mathcal{A}'}$ are everywhere transverse. The rest of the statement readily follows.\n\\end{proof}\n\\subsection{Lie subalgebroids}\n\\label{sec:liesubalgds}\nIn this section we consider subobjects of Lie algebroids. Let $\\mathcal{A} \\to X$ be a Lie algebroid and $(\\varphi,f)\\colon (\\mathcal{B}, N) \\hookrightarrow (\\mathcal{A},X)$ be a vector subbundle supported on $N \\subseteq X$. It is an \\emph{anchored subbundle} of $\\mathcal{A}$ supported on $N$ if $\\varphi$ is an anchored morphism, in which case $\\rho_{B} = \\rho_{A}|_N$ restricted to $\\mathcal{B}$. It is easy to see that if $(\\rho_\\mathcal{A}|_N)(\\mathcal{B}) \\subseteq TN$ then $\\mathcal{B}$ carries a unique anchored bundle structure for which $\\varphi$ is such an anchored morphism.\n\nNext, consider the submodule $\\Gamma(\\mathcal{A},\\mathcal{B}) := \\{v \\in \\Gamma(\\mathcal{A}) \\, | \\, v|_N \\in \\Gamma(\\mathcal{B})\\}$ of $\\Gamma(\\mathcal{A})$, which comes equipped with a surjective restriction map $\\Gamma(\\mathcal{A},\\mathcal{B}) \\to \\Gamma(\\mathcal{B})$ with kernel $\\Gamma(\\mathcal{A},0_N)$. Note that $\\mathcal{B}$ is a \\emph{Lie subalgebroid} of $\\mathcal{A}$ supported on $N$ if $\\varphi$ is a Lie algebroid morphism, in which case the restriction map $\\Gamma(\\mathcal{A},\\mathcal{B}) \\to \\Gamma(\\mathcal{B})$ is a Lie algebra homomorphism. The following proposition is noted in \\cite[Proposition 2.15]{Meinrenken17} and in part in \\cite[Remark 2.1.21]{Li13}.\n\\begin{prop}[\\cite{Meinrenken17}]\\label{prop:liesubalgeroid} Let $(\\varphi,f)\\colon (\\mathcal{B},N) \\hookrightarrow (\\mathcal{A},X)$ be an anchored subbundle supported on $N$. Then the following are equivalent:\n\t%\n\t\\begin{itemize}\n\t\\item The submodule $\\Gamma(\\mathcal{A},\\mathcal{B}) \\subseteq \\Gamma(\\mathcal{A})$ is involutive;\n\t\\item $\\mathcal{B}$ carries a unique Lie algebroid structure for which $\\varphi$ is a Lie algebroid morphism.\n\t\\end{itemize}\n\\end{prop}\n\\bp Assume that $\\Gamma(\\mathcal{A},\\mathcal{B})$ is involutive. We show that $\\Gamma(\\mathcal{A},0_N)$ is an ideal in $\\Gamma(\\mathcal{A},\\mathcal{B})$, i.e.\\\n\\begin{equation*}\n\t[\\Gamma(\\mathcal{A},0_N), \\Gamma(\\mathcal{A},\\mathcal{B})]_\\mathcal{A} \\subseteq \\Gamma(\\mathcal{A},\\mathcal{B}).\n\\end{equation*}\nLet $v \\in \\Gamma(\\mathcal{A},0_N)$. Then $v = \\sum_i f_i v_i$ for some $v_i \\in \\Gamma(\\mathcal{A})$ and $f_i \\in I_N$. Given $w \\in \\Gamma(\\mathcal{A},\\mathcal{B})$, we have $\\rho_\\mathcal{A}(w)|_N \\in \\Gamma(TN)$. Consequently $\\mathcal{L}_{\\rho_\\mathcal{A}(w)} f_i|_N = 0$ for all $i$, and the Leibniz rule gives that $[v,w]_\\mathcal{A}|_N = 0$, as we have\n\\begin{equation*}\n\t[v, w]_\\mathcal{A} = [\\sum_i f_i v_i, w]_\\mathcal{A} = \\sum_i (f_i [v_i,w]_{\\mathcal{A}} - \\mathcal{L}_{\\rho_\\mathcal{A}(w)} f_i \\cdot v_i).\n\\end{equation*}\nThus $\\Gamma(\\mathcal{B}) \\cong \\Gamma(\\mathcal{A},\\mathcal{B}) \/ \\Gamma(\\mathcal{A},0_N)$ inherits a Lie bracket, making $\\mathcal{B}$ into a Lie algebroid, since the Leibniz rule for $\\rho_{\\mathcal{A}}$ implies the Leibniz rule is satisfied for $\\rho_{\\mathcal{B}}$. The converse is immediate.\n\\end{proof}\n\\begin{rem}\\label{rem:liesubalgd} Sometimes one includes in the definition, apart from the conditions that $(\\rho_{A}|_N)(\\mathcal{B}) \\subseteq TN$ and involutivity of $\\Gamma(\\mathcal{A},\\mathcal{B})$, the condition that $\\Gamma(\\mathcal{A},0_N)$ is an ideal in $\\Gamma(\\mathcal{A},\\mathcal{B})$. \\autoref{prop:liesubalgeroid} shows this latter condition is a consequence of the first two.\n\\end{rem}\nFor later reference we recall the following special examples of Lie subalgebroids.\n\\begin{defn} A Lie subalgebroid $(\\varphi,f)\\colon (\\mathcal{B},X') \\to (\\mathcal{A},X)$ is \\emph{wide} if $X' = X$ and $f = {\\rm id}_X$.\n\\end{defn}\n\\begin{exa}\\label{exa:regfoliation} Let $\\mathcal{F}$ be a regular foliation on $X$ and $D = T \\mathcal{F} \\subseteq TX$ the associated involutive distribution. Then $\\mathcal{A}_\\mathcal{F} := D$ is a Lie subalgebroid of $TX$. Moreover, if $\\mathcal{A} \\to X$ is a regular Lie algebroid with injective anchor, then $\\mathcal{A} = \\mathcal{A}_\\mathcal{F}$ for $\\mathcal{F}$ associated to the distribution $D = \\rho_\\mathcal{A}(\\mathcal{A})$.\n\\end{exa}\n\\begin{exa}\\label{exa:adistribution} Let $\\mathcal{A} \\to X$ be a Lie algebroid with an \\emph{$\\mathcal{A}$-distribution}, i.e.\\ a smooth subbundle $D_\\mathcal{A} \\subseteq \\mathcal{A}$. Its image $D := \\rho_\\mathcal{A}(D_\\mathcal{A}) \\subseteq TX$, need not be a distribution. If $D_\\mathcal{A} \\subseteq \\mathcal{A}$ is $\\mathcal{A}$-involutive (closed under $[\\cdot,\\cdot]_\\mathcal{A}$), then it is an $\\mathcal{A}$-Lie subalgebroid with $\\mathcal{A}$-anchor the natural inclusion.\n\\end{exa}\n\\subsection{$\\mathcal{A}$-invariant submanifolds}\n\\label{sec:degenloci}\nIn this section we discuss $\\mathcal{A}$-invariant submanifolds, i.e.\\ when one can restrict a given Lie algebroid, as a Lie algebroid. We follow in part \\cite{Pym13}.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid. Then a submanifold $N \\subseteq X$ is \\emph{$\\mathcal{A}$-invariant} if $\\rho_\\mathcal{A}$ is tangent to $N$, i.e.\\ $\\rho_\\mathcal{A}|_N$ has image inside $TN$ (instead of merely $TX|_N$).\n\\end{defn}\nFor closed submanifolds $N \\subseteq X$, a vector field $V \\in \\Gamma(TX)$ preserves its vanishing ideal $I_N$ if and only if it is tangent to $N$. Hence a closed submanifold $N$ is $\\mathcal{A}$-invariant if and only if $I_N$ is preserved by $\\mathcal{A}$, i.e.\\ we have $\\mathcal{L}_{\\rho_\\mathcal{A}(v)} I_N \\subseteq I_N$ for all $v \\in \\Gamma(\\mathcal{A})$. Thus $\\mathcal{A}$-invariant submanifolds consist of orbits of $\\mathcal{A}$, i.e.\\ are unions of (opens of) leaves of the singular foliation given by $\\rho_\\mathcal{A}$.\n\\begin{prop}\\label{prop:ainvariantrestr} Let $\\mathcal{A}$ be a Lie algebroid and $N$ be an $\\mathcal{A}$-invariant submanifold of $X$. Then $(\\mathcal{A}|_N, \\rho_\\mathcal{A}|_N) \\to N$ with restriction of $[\\cdot,\\cdot]_\\mathcal{A}$ is a Lie subalgebroid of $\\mathcal{A}$ along $N$.\n\\end{prop}\n\\bp Using \\autoref{prop:liesubalgeroid}, because $\\mathcal{A}|_N \\to TN$ is an anchored bundle due to $\\mathcal{A}$-invariance of $N$, we need only show that $\\Gamma(\\mathcal{A},\\mathcal{A}|_N)$ is involutive. However, this is immediate.\n\\end{proof}\n\\begin{rem}\\label{rem:inverseimagesmooth} The proof of \\autoref{prop:ainvariantrestr} shows\nmore generally, given\na submanifold $\\iota\\colon N \\hookrightarrow X$, that $\\iota^! \\mathcal{A} := \\rho_\\mathcal{A}^{-1}(TN)$ is a Lie subalgebroid of $\\mathcal{A}$ along $N$ if it is a smooth subbundle of $\\mathcal{A}|_N$ (see \\cite[Proposition 2.17]{Meinrenken17}). Note here that $\\iota^! \\mathcal{A} = \\mathcal{A}|_N$ if and only if $N$ is $\\mathcal{A}$-invariant.\n\\end{rem}\nThe set of $\\mathcal{A}$-invariant closed submanifolds of $X$ admits unions and intersections. We say a subset of $X$ is \\emph{$\\mathcal{A}$-invariant} if it is a union of $\\mathcal{A}$-invariant submanifolds.\n\\begin{lem}\\label{lem:ainvariance} Let $N$ and $N'$ be $\\mathcal{A}$-invariant closed submanifolds of $X$. Then their union $N \\cup N'$ and intersection $N \\cap N'$ are $\\mathcal{A}$-invariant closed subsets.\n\\end{lem}\n\\bp The sets $N \\cup N'$ and $N \\cap N'$ have vanishing ideals given by $I_{N \\cup N'} = I_N \\cap I_{N'} = I_N \\cdot I_{N'}$ and $I_{N \\cap N'} = I_N + I_{N'}$. Using the product rule $\\mathcal{L}_{\\rho_\\mathcal{A}(v)}(f g) = (\\mathcal{L}_{\\rho_\\mathcal{A}(v)}f) \\cdot g + f \\cdot (\\mathcal{L}_{\\rho_{\\mathcal{A}}(v)}g)$, we conclude that $I_{N \\cup N'}$ is $\\mathcal{A}$-invariant. The $\\mathcal{A}$-invariance of $I_{N \\cap N'}$ is immediate by additivity. Alternatively, one can note that $\\mathcal{A}$-invariant submanifolds are precisely unions of $\\mathcal{A}$-orbits.\n\\end{proof}\n\\begin{rem}\\label{rem:preservingideals} The proof of \\autoref{lem:ainvariance} shows that more generally, given a collection of vector fields $\\mathcal{F} \\subseteq \\Gamma(TX)$ preserving two ideals $I$, $I'$, i.e.\\ such that $\\mathcal{L}_v I \\subseteq I$ and $\\mathcal{L}_v I' \\subseteq I'$ for all $v \\in \\mathcal{F}$, the collection $\\mathcal{F}$ also preserves $I + I'$ and $I \\cap I'$ (see \\cite[Proposition 2.1.10]{Pym13}). In \\autoref{lem:ainvariance} this is applied to the collection $\\mathcal{F} := \\rho_{A}(\\Gamma(\\mathcal{A}))$ and ideals $I := I_N$, $I' := I_{N'}$.\n\\end{rem}\nWhen considering residue maps associated to $\\mathcal{A}$-invariant submanifolds in \\autoref{sec:residuemaps}, we will need the following terminology, concerning properties of the restricted anchor map.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $N \\subseteq X$ an $\\mathcal{A}$-invariant submanifold. Then $N$ is \\emph{projective} if the restriction $\\mathcal{A}|_N$ is projective, and \\emph{transitive} if $\\mathcal{A}|_N$ is transitive.\n\\end{defn}\nCertainly any transitive $\\mathcal{A}$-invariant submanifold is in particular projective. Any orbit $\\mathcal{O}$ of $\\mathcal{A}$ is a transitive $\\mathcal{A}$-invariant submanifold. Further examples will be given later in \\autoref{sec:residues}.\n\nGiven a projective $\\mathcal{A}$-invariant submanifold $N \\subseteq X$, by the definition of projectivity, the Serre--Swan theorem provides a Lie algebroid $\\mathcal{B} \\to N$ satisfying $\\Gamma(\\mathcal{B}) \\cong (\\rho_A|_N)(\\Gamma(\\mathcal{A}|_N))$. There is an induced Lie algebroid morphism $\\widetilde{\\rho}_\\mathcal{A} \\colon \\mathcal{A}|_N \\to \\mathcal{B}$ obtained from the natural map on sections. This fits in a short exact sequence of Lie algebroids over $N$, namely\n\\begin{equation*}\n\t0 \\to \\ker \\widetilde{\\rho}_\\mathcal{A} \\to \\mathcal{A}|_N \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nAs we did in \\autoref{sec:liealgebroids}, we call $\\ker \\widetilde{\\rho}_\\mathcal{A}$ the (bundle of) \\emph{germinal isotropy} of $\\mathcal{A}|_N$, after \\cite{AndroulidakisZambon17}. \n\\subsection{Degeneracy loci}\nIn this section we define the degeneracy loci of $\\mathcal{A}$ and show that they are $\\mathcal{A}$-invariant subsets. Given an integer $k \\geq 0$, the anchor of $\\mathcal{A}$ defines a map $\\wedge^k \\rho_\\mathcal{A}\\colon \\wedge^k \\mathcal{A} \\to \\wedge^k TX$. This induces a map $\\wedge^k \\rho_\\mathcal{A}\\colon \\Gamma(\\wedge^k \\mathcal{A} \\otimes \\wedge^k T^*X) \\to C^\\infty(X)$, whose image is an ideal $I_{\\mathcal{A},k}$.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid. The $k$th \\emph{degeneracy locus} of $\\mathcal{A}$ is the subspace $X_{\\mathcal{A},k} \\subseteq X$ defined by the ideal $I_{\\mathcal{A},k+1} \\subseteq C^\\infty(X)$, where the rank of $\\rho_\\mathcal{A}$ is $k$ or less.\n\\end{defn}\n\\begin{prop}[{\\cite[Proposition 2.2.9]{Pym13}}]\\label{prop:degenlocus} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $k \\geq 0$. Then $X_{\\mathcal{A},k}$ is an $\\mathcal{A}$-invariant submanifold whenever it is a smooth submanifold.\n\\end{prop}\n\\bp Assume that $X_{\\mathcal{A},k}$ is smooth and denote the map $\\wedge^{k+1} \\rho_\\mathcal{A}$ by $\\varphi_{k}\\colon \\wedge^{k+1} \\mathcal{A} \\to \\wedge^{k+1} TX$. Let $v \\in \\Gamma(\\mathcal{A})$ and $f \\in I_{\\mathcal{A},k}$. Then by definition there exists $\\xi \\in \\Gamma(\\wedge^{k+1} \\mathcal{A})$ and $\\omega \\in \\Omega^{k+1}(X)$ such that $f = \\langle \\varphi_k(\\xi), \\omega \\rangle$. Using the compatibility of the bracket and anchor we compute that\n\\begin{align*}\n\t\\mathcal{L}_{\\rho_\\mathcal{A}(v)} f &= \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\langle \\varphi_k(\\xi), \\omega \\rangle = \\langle \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\varphi_k(\\xi), \\omega \\rangle + \\langle \\varphi_k(\\xi), \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\omega \\rangle\\\\\n\t&= \\langle \\varphi_k(\\mathcal{L}_{v} \\xi), \\omega\\rangle + \\langle \\varphi_k(\\xi), \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\omega \\rangle \\in I_{\\mathcal{A},k}.\n\\end{align*}\nAs $I_{\\mathcal{A},k}$ is preserved under the action of $\\mathcal{A}$, we conclude that $X_{\\mathcal{A},k}$ is $\\mathcal{A}$-invariant.\n\\end{proof}\n\\begin{rem}\\label{exa:denseisolocus} Let $\\mathcal{A}^n \\to X^n$ be a Lie algebroid of divisor-type. Then $X \\backslash X_\\mathcal{A} = Z_\\mathcal{A} = X_{\\mathcal{A},n-1}$. In fact, we have an equality of ideals $I_\\mathcal{A} = I_{\\mathcal{A},n}$. Both $X_\\mathcal{A}$ and $Z_\\mathcal{A}$ are $\\mathcal{A}$-invariant subsets.\n\\end{rem}\n\\subsection{Singular Lie subalgebroids}\n\\label{sec:singularliealgebroids}\nIn this section we take a slight step into the world of singular foliations. While those underlying the Lie algebroids of interest to us are relatively mild, it is useful to relate our discussion to the language in existing literature, particularly \\cite{AndroulidakisSkandalis09, AndroulidakisZambon13, AndroulidakisZambon16, AndroulidakisZambon17, LaurentGengouxLavauStrobl18, Zambon18}. To extend the results in this paper to more singular Poisson structures, one is forced to work with sheaves or modules which no longer have an underlying bundle. Because we defer this to future work, we will be somewhat brief here.\n\\begin{rem} As a general comment, projective $C^\\infty(X)$-modules are not necessarily locally finitely-generated, i.e.\\ need not necessarily correspond to section modules of vector bundles. However, our use of the term `projective' will presuppose that the module in question is locally finitely-generated. Indeed, for us such modules are often obtained as images of bundle maps.\n\\end{rem}\nLet $X$ be a manifold and $\\mathcal{C}^\\infty_X$ be the sheaf of smooth functions on $X$, with $C^\\infty(X) = \\mathcal{C}^\\infty_X(X)$. Throughout this section, a \\emph{module} will refer to a $C^\\infty(X)$-module without further mention.\n\nLet $E \\to X$ be a vector bundle and $\\mathcal{E}' \\subseteq \\mathcal{E} := \\Gamma(E)$ be a submodule. Given a smooth map $f \\colon X' \\to X$, let $f^* E \\to X'$ be the pullback bundle. The \\emph{pullback module} $f^* \\mathcal{E}'$ of $\\mathcal{E}'$ along $f$ is\n\\begin{equation*}\n\tf^* \\mathcal{E}' := \\langle g f^* \\xi \\, | \\, g \\in C^\\infty(X'), \\xi \\in \\mathcal{E}' \\rangle \\subseteq \\Gamma(X'; f^*E).\n\\end{equation*}\nIf $\\iota_{X'}\\colon X \\hookrightarrow X$ is a submanifold, we call $\\iota_{X'}^* \\mathcal{E}'$ the \\emph{restriction} of $\\mathcal{E}'$ to $X'$.\nA module is \\emph{finitely generated} if there exist global sections $\\xi_i$ of $\\mathcal{E}$ such that $\\mathcal{E} = C^\\infty(X) \\langle \\xi_i \\rangle$, and \\emph{locally finitely generated} if there exists an open cover $\\{U_i\\}$ of $X$ such that each restriction $i_{U_i}^*\\mathcal{E}$ is finitely generated. Call a module $\\mathcal{E}$ of \\emph{vector bundle-type} when $\\mathcal{E} \\cong \\Gamma(E)$ for a vector bundle $E \\to X$.\n\\begin{defn} A module $\\mathcal{E}$ is \\emph{projective} if it has a global basis of generating sections, or, equivalently, if it is a direct summand of a finitely generated free module. \n\\end{defn}\n\\begin{rem} There is a one-to-one correspondence between locally finitely generated projective modules $\\mathcal{E}$, and locally free sheaves of $\\mathcal{C}^\\infty_X$-modules $\\{U \\mapsto \\mathcal{E}_U\\}$. This extends to singular foliations (adding involutivity and possibly dropping projectiveness), giving a one-to-one correspondence between singular foliations and involutive sheaves of locally finitely generated submodules of $\\mathfrak{X}_X$ \\cite{Wang17}. Adding projectivity (or local freeness), we obtain that projective singular foliations are exactly locally free sheaves of involutive submodules of $\\mathfrak{X}_X$.\n\\end{rem}\nThe \\emph{Serre--Swan correspondence} \\cite{Serre55, Swan62} gives an equivalence of categories between finitely generated projective modules $\\mathcal{E}$, and vector bundles $E \\to X$ via the assignment $\\mathcal{E} \\mapsto \\Gamma(E)$. In other words, locally finitely generated projective modules are of vector bundle-type.\n\nFocusing on involutivity, using a given Lie algebroid $\\mathcal{A} \\to X$, results in the following (we give the definition in terms of sheaves, instead of using compactly-supported sections).\n\\begin{defn}[c.f.\\ \\cite{Zambon18}]\\label{defn:singularliealgebroid} A \\emph{singular subbundle} of $\\mathcal{A}$ is a sheaf of locally finitely generated submodules of its sheaf of sections $\\Gamma(\\mathcal{A})$.  A \\emph{singular Lie subalgebroid} of $\\mathcal{A}$ is an involutive singular subbundle. We also use the associated module of global sections interchangeably.\n\\end{defn}\n\\begin{exa}\tA wide Lie subalgebroid $\\mathcal{A}' \\subseteq \\mathcal{A}$ induces a singular Lie subalgebroid $\\Gamma(\\mathcal{A}')$.\n\\end{exa}\n\\begin{exa} \\autoref{prop:liesubalgeroid} shows that given a Lie subalgebroid $\\mathcal{B}$, the module $\\Gamma(\\mathcal{A},\\mathcal{B})$ is a singular Lie subalgebroid of $\\mathcal{A}$. It is not necessarily projective (see also \\autoref{sec:elementarymodifications}).\n\\end{exa}\n\nA projective singular Lie subalgebroid of $\\mathcal{A}$ defines an $\\mathcal{A}$-Lie algebroid $\\mathcal{A}'$ along with a Lie algebroid morphism $\\mathcal{A}' \\to \\mathcal{A}$. Note however that $\\mathcal{A}'$ is generally \\emph{not} a Lie subalgebroid of $\\mathcal{A}$.\n\nSingular Lie subalgebroids of $TX$ are called \\emph{singular foliations} in the sense of Androulidakis--Skandalis \\cite{AndroulidakisSkandalis09}. Moreover, projective singular foliations $\\mathcal{F}$ are also called \\emph{almost-regular} \\cite{Debord01}, and give rise to almost-injective Lie algebroids $\\mathcal{A}_\\mathcal{F}$ with $\\Gamma(\\mathcal{A}_\\mathcal{F}) \\cong \\mathcal{F}$. Note that underlying any singular Lie subalgebroid $\\mathcal{F}_\\mathcal{A}$ of $\\mathcal{A}$ there is a singular foliation $\\mathcal{F} := \\rho_{A}(\\mathcal{F}_\\mathcal{A})$. However, in general there is no relation between projectivity of the modules $\\mathcal{F}_\\mathcal{A} \\subseteq \\Gamma(\\mathcal{A})$ and $\\mathcal{F} \\subseteq \\Gamma(TX)$.\n\\subsubsection{On projectivity}\nEach submodule $\\mathcal{E}'$ of a projective module $\\mathcal{E}$ has an evaluation map ${\\rm ev}\\colon X \\times \\mathcal{E}' \\to E$ defined by ${\\rm ev}(x,\\xi) := \\xi(x)$. Define ${\\rm ev}_x\\colon \\mathcal{E}' \\to E_x$ by ${\\rm ev}_x := {\\rm ev}(x, \\cdot)$ for $x \\in X$. We will denote by $E_x' := {\\rm ev}_x(\\mathcal{E}')$ the image of this map, which is a vector subspace of the fiber $E_x$ of $E$. Given a point $x \\in X$, let $I_x \\subseteq C^\\infty(X)$ be its vanishing ideal. Moreover, given a submodule $\\mathcal{E}'$ of $\\mathcal{E} = \\Gamma(E)$, consider the quotient vector space $\\mathcal{E}'_x := \\mathcal{E}' \/ I_x \\mathcal{E}'$ of $\\mathcal{E}'$ by $I_x$, called the \\emph{fiber} of $\\mathcal{E}'$ at $x$. Note that $\\mathcal{E}'_x = E'_x$ when $\\mathcal{E}'$ is of vector bundle-type. We have:\n\\begin{prop} Let $\\mathcal{E}' \\subseteq \\mathcal{E}$ be a submodule on $X$ where $\\mathcal{E} = \\Gamma(E)$ is of vector bundle-type. Then $x \\mapsto \\dim E'_x$ is lower semi-continuous while $x \\mapsto \\dim \\mathcal{E}'_x$ is upper semi-continuous.\n\\end{prop}\nWhen $\\mathcal{E}$ is locally finitely generated, the vector spaces $\\mathcal{E}_x$ are finite-dimensional for all $x \\in X$ (see \\cite{AndroulidakisSkandalis09}). The evaluation map ${\\rm ev}_x$ vanishes on $I_x \\mathcal{E}'$, so that it descends to a surjective homomorphism $\\widetilde{\\rm ev}_x\\colon \\mathcal{E}'_x \\to E'_x$. When $\\mathcal{E}$ is a Lie module and $\\mathcal{E}' \\subseteq \\mathcal{E}$ an involutive submodule, the kernel of ${\\rm ev}_x$ is a Lie subalgebra of $\\mathcal{E}'$, and $I_x \\mathcal{E}'$ is an ideal in this subalgebra. This results in the conclusion that $\\mathfrak{a}_x := \\ker {\\rm ev}_x \/ I_x \\mathcal{E}' = \\ker \\widetilde{\\rm ev}_x$ is a Lie algebra, called the \\emph{isotropy} of $\\mathcal{E}'$ at $x \\in X$ when $\\mathcal{E} = \\Gamma(TX)$. There is no induced bracket on $\\mathcal{E}'_x$. There is a short exact sequence\n\\begin{equation*}\n0 \\to \\mathfrak{a}_x \\to \\mathcal{E}'_x \\stackrel{\\widetilde{\\rm ev}_x}{\\to} E_x' \\to 0.\n\\end{equation*}\n\\begin{prop}[{\\cite[Lemma 1.3]{AndroulidakisZambon13}}] Let $\\mathcal{E}' \\subseteq \\mathcal{E}$ be a submodule on $X$ where $\\mathcal{E} = \\Gamma(E)$ is of vector bundle-type. Then $\\dim \\mathfrak{g}_x$ is bounded from above by ${\\rm rank}(E)$.\n\\end{prop}\nThe Serre--Swan correspondence implies that a locally finitely-generated module $\\mathcal{F}$ is projective if and only if the dimension of $\\mathcal{F}_x$ is constant for all $x \\in X$. We can pinpoint the failure of projectivity more precisely, following \\cite{AndroulidakisZambon17}. Let $\\mathcal{F}_\\mathcal{A}$ be a singular Lie subalgebroid of $\\mathcal{A}$. Then the module $\\Gamma(\\mathcal{A})$ is projective, and $\\mathcal{F}_\\mathcal{A}$ is a submodule of it. As such the discussion above leads to a short exact sequence for each point $x \\in X$, namely\n\\begin{equation*}\n\t0 \\to \\mathfrak{a}_x \\to \\mathcal{F}_{\\mathcal{A},x} \\to \\mathcal{A}_{\\mathcal{F}_\\mathcal{A},x} \\to 0,\n\\end{equation*}\nwhere $\\mathcal{A}_{\\mathcal{F}_\\mathcal{A},x} = {\\rm ev}_x(\\mathcal{F}_\\mathcal{A}) \\subseteq \\mathcal{A}_x$ is the evaluation, and $\\mathcal{F}_{\\mathcal{A},x} = \\mathcal{F}_\\mathcal{A} \/ I_x \\mathcal{F}_\\mathcal{A}$. Now assume that $\\mathcal{F}_\\mathcal{A}$ is induced by an $\\mathcal{A}$-Lie algebroid. In other words, assume that  we have $\\varphi_{\\mathcal{A}}\\colon \\mathcal{A}' \\to \\mathcal{A}$ with $\\mathcal{F}_\\mathcal{A} = \\varphi_{\\mathcal{A}}(\\Gamma(\\mathcal{A}')) \\subseteq \\Gamma(\\mathcal{A})$. We then obtain from $\\varphi_{\\mathcal{A}}$ another short exact sequence, namely\n\\begin{equation*}\n\t0 \\to \\mathfrak{g}_x \\to \\mathcal{A}'_x \\to \\mathcal{A}_{\\mathcal{F}_\\mathcal{A},x} \\to 0.\n\\end{equation*}\nThe $\\mathcal{A}$-anchor $\\varphi_{\\mathcal{A}}\\colon \\Gamma(\\mathcal{A}') \\to \\Gamma(\\mathcal{A})$ on sections gives another short exact sequence:\n\\begin{equation*}\n\t0 \\to \\mathfrak{h}_x \\to \\mathcal{A}'_x \\to \\mathcal{F}_{\\mathcal{A},x} \\to 0.\n\\end{equation*}\nWe call the kernel $\\mathfrak{h}_x$ the \\emph{germinal $\\mathcal{A}$-isotropy} of $\\mathcal{F}_\\mathcal{A}$ at $x \\in X$. If the module $\\mathcal{F}_\\mathcal{A}$ is projective, so that $\\mathcal{F}_\\mathcal{A} \\cong \\Gamma(\\mathcal{B})$ for $\\mathcal{B} \\to X$, there is an induced Lie algebroid morphism $\\widetilde{\\varphi}_\\mathcal{A}\\colon \\mathcal{A}' \\to \\mathcal{B}$, and then $\\mathfrak{h}_x = \\ker(\\widetilde{\\varphi}_{\\mathcal{A},x}) \\subseteq \\mathcal{A}'_x$ as in \\autoref{defn:germinalisotropy} for $\\mathcal{A} = TX$. The above sequences are related, as discussed in \\cite[Section 1.3]{AndroulidakisZambon17} for the case of $\\mathcal{A} = TX$. Indeed, we have a short exact sequence\n\\begin{equation*}\n\t0 \\to \\mathfrak{h}_x \\to \\mathfrak{g}_x \\to \\mathfrak{a}_x \\to 0.\n\\end{equation*}\nIn conclusion, projectivity of $\\mathcal{F}_\\mathcal{A}$ is equivalent to $\\dim(\\mathfrak{h}_x)$ being constant for all $x \\in X$. This criterion for projectivity also follows from the fact that the above Lie algebroid morphism $\\widetilde{\\varphi}_\\mathcal{A}$ is surjective by construction, so that it induces a Lie algebroid extension sequence\n\\begin{equation*}\n\t0 \\to \\ker(\\widetilde{\\varphi}_\\mathcal{A}) \\to \\mathcal{A}' \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nIn \\cite{LaurentGengouxLavauStrobl18} the authors discuss the notion of a \\emph{projective resolution} of a singular foliation $\\mathcal{F}$, which is an exact sequence of sheaves of $C^\\infty(X)$-modules:\n\\begin{equation*}\n\t\\dots \\to \\Gamma(E_{-2}) \\to \\Gamma(E_{-1}) \\to \\mathcal{F} \\to 0,\n\\end{equation*}\nwhere $E_{-i}$ for $i \\geq 1$ are vector bundles over $X$, with anchor map $E_{-1} \\to TX$ with image $\\mathcal{F}$.\nIt is clear that this definition makes sense for singular Lie subalgebroids as well upon replacing $TX$ by $\\mathcal{A}$ (giving \\emph{projective $\\mathcal{A}$-resolutions}). The \\emph{projective $\\mathcal{A}$-dimension} of such a module (or sheaf of submodules) is then defined as the minimal length of a projective resolution, minus one. Thus projective singular Lie subalgebroids are those with zero projective $\\mathcal{A}$-dimension. In future work we will treat situations of positive projective dimension.\n\\subsection{Ideal Lie algebroids}\n\\label{sec:idealliealgebroids}\nIn this section we discuss what we call ideal Lie algebroids, which are Lie algebroids defined using a given ideal of functions. Of particular importance are those created using divisor ideals. Let $\\mathcal{A} \\to X$ be a Lie algebroid and let $I \\subseteq C^\\infty(X)$ be an ideal. Consider $I \\cdot \\Gamma(\\mathcal{A}) \\subseteq \\Gamma(\\mathcal{A})$ consisting of all finite sums of products of elements of $I$ and $\\Gamma(\\mathcal{A})$, and let $\\Gamma(\\mathcal{A})_I := \\{v \\in \\Gamma(\\mathcal{A}) \\, | \\, \\mathcal{L}_{\\rho_{A}(v)} I \\subseteq I\\}$ be those sections preserving $I$. Both are $C^\\infty(X)$-submodules of $\\Gamma(\\mathcal{A})$. In this section we sometimes shorten $\\mathcal{L}_{\\rho_{\\mathcal{A}}(v)}$ to $\\mathcal{L}_v$.\n\\begin{lem}\\label{lem:liesubalgs} Let $I \\subseteq C^\\infty(X)$ be an ideal. Then $I \\cdot \\Gamma(\\mathcal{A}) \\subseteq \\Gamma(\\mathcal{A})_I \\subseteq \\Gamma(\\mathcal{A})$ are Lie subalgebras.\n\\end{lem}\n\\bp Let $v, w \\in \\Gamma(\\mathcal{A})_I$ and $f \\in I$. Then $\\mathcal{L}_v f = g$ and $\\mathcal{L}_w f = h$ for $g, h \\in I$. Hence\n\\begin{equation*}\n\t\\mathcal{L}_{[v,w]} f = (\\mathcal{L}_v \\mathcal{L}_w - \\mathcal{L}_w \\mathcal{L}_v) f = \\mathcal{L}_v h - \\mathcal{L}_w g.\n\\end{equation*}\nAs $g, h \\in I$ we have $\\mathcal{L}_{[v,w]} f \\in I$ for all $f \\in I$, hence $[v,w] \\in \\Gamma(\\mathcal{A})_I$.\n\nLet $v, w \\in I \\cdot \\Gamma(\\mathcal{A})$ with $v = \\sum_i f_i v_i$ and $w = \\sum_j f'_j w_j$ for some $f_i, f'_j \\in I$ and $v_i, w_j \\in \\Gamma(\\mathcal{A})$. Then we have $[v,w] \\in I \\cdot \\Gamma(\\mathcal{A})$ by the Leibniz rule as each term starts with an element of $I$,\n\\begin{equation*} \n[v,w] = \\textstyle{\\sum}_{i,j} [f_i v_i, f'_j w_j] = \\textstyle{\\sum}_{i,j} \\left( f_i f'_j [v_i, w_j] + f_i (\\mathcal{L}_{v_i} f'_j) \\cdot w_j - f'_j (\\mathcal{L}_{w_j} f_i) \\cdot v_i\\right).\n\\end{equation*}\nLet $v \\in I \\cdot \\Gamma(\\mathcal{A})$. Then $v = \\sum_i f_i v_i$ for $f_i \\in I$ and $v_i \\in \\Gamma(\\mathcal{A})$. For $f \\in I$ we have $\\mathcal{L}_v f = \\sum_i f_i (\\mathcal{L}_{v_i} f)$. As $f_i \\in I$ for all $i$ and $I$ is an ideal, we conclude that $\\mathcal{L}_v f \\in I$ so that $v \\in \\Gamma(\\mathcal{A})_I$ as desired.\n\\end{proof}\nIt is immediate that if $I$ is locally finitely generated, then both $I \\cdot \\Gamma(\\mathcal{A})$ and $\\Gamma(\\mathcal{A})_I$ are as well. By \\autoref{lem:liesubalgs} they are then both singular Lie subalgebroids of $\\mathcal{A}$. Consequently, if they are also projective, they define $\\mathcal{A}$-Lie algebroids by the Serre--Swan correspondence. Due to \\autoref{lem:liesubalgs}, there is then a sequence of Lie algebroid morphisms as follows:\n\\begin{equation*}\n\tI \\cdot \\mathcal{A} \\to \\mathcal{A}_I \\to \\mathcal{A} \\to TX.\n\\end{equation*}\nWe will mostly focus on the projective case, i.e.\\ when $\\Gamma(\\mathcal{A})_I$ and $I\\cdot \\Gamma(\\mathcal{A})$ are projective.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $I \\subseteq C^\\infty(X)$ be an ideal. The primary and secondary \\emph{ideal Lie algebroids} associated to $(\\mathcal{A},I)$ are the Lie algebroids $\\mathcal{A}_I \\to X$ such that $\\Gamma(\\mathcal{A}_I) \\cong \\Gamma(\\mathcal{A})_I$, and $I \\cdot \\mathcal{A} \\to X$ such that $\\Gamma(I \\cdot \\mathcal{A}) \\cong I \\cdot \\Gamma(\\mathcal{A})$.\n\\end{defn}\nBoth ideal Lie algebroids are unique -- provided they exist -- and their $\\mathcal{A}$-anchors are the natural inclusions on sections. However, their $\\mathcal{A}$-anchors are not injective when viewed as bundle maps, unless $I$ is trivial. The isomorphism locus of the $\\mathcal{A}$-anchor of $\\mathcal{A}_I$, and of $I \\cdot \\mathcal{A}$, as well as the morphism between $I\\cdot \\mathcal{A}$ and $\\mathcal{A}_I$, is given by $X \\backslash Z_I$, the complement of the support of $I$. In other words, these three Lie algebroid morphisms are of divisor-type supported on $Z_I$, which in particular means that the ideal Lie algebroids $\\mathcal{A}_I$ and $I \\cdot \\mathcal{A}$ are both of $\\mathcal{A}$-divisor-type.\n\nOur main interest lies in the case where $\\mathcal{A} = TX$ and $I$ is a divisor ideal. Then both ideal Lie algebroids are of divisor-type, with divisors ${\\rm div}(I \\cdot TX)$ and ${\\rm div}(TX_I)$ supported on $Z_I$, and $I_{I \\cdot TX} = I^n$ if $\\dim X = n$ (and similarly using $\\mathcal{A}$). We introduce the following terminology.\n\\begin{defn}\\label{defn:divprojstandard} A divisor ideal $I \\subseteq C^\\infty(X)$ is said to be \\emph{projective} if $\\Gamma(TX)_I$ is projective. A divisor ideal is \\emph{standard} if it is projective and the divisor ideal of $TX_I$ satisfies $I_{TX_I} = I$.\n\\end{defn}\nNot every divisor ideal is projective, nor is every projective divisor ideal standard. Projectivity of $I \\cdot \\Gamma(A)$ is guaranteed for divisor ideals, as is also noted in \\cite[Lemma 1.11]{Nistor15}.\n\\begin{prop}\\label{prop:dividealalgebroid} Given a Lie algebroid $\\mathcal{A}^n \\to X$ and divisor ideal $I$, $I \\cdot \\Gamma(\\mathcal{A})$ is projective.\n\\end{prop}\n\\bp \nThere is a short exact sequence of sheaves $0 \\to I \\cdot \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A}) \/ I \\cdot \\Gamma(\\mathcal{A}) \\to 0$. As both $\\Gamma(\\mathcal{A})$ and $I$ are locally free, so is $\\Gamma(\\mathcal{A}) \/ I \\cdot \\Gamma(\\mathcal{A}) \\cong (I^{-1})^n$ (the inverse of $I$ as a sheaf). The same then holds for $I \\cdot \\Gamma(\\mathcal{A})$ as any short exact sequence in a projective module splits.\n\\end{proof}\nWhen $I$ is not necessarily standard, we expect the following relation between $I$ and $I_{TX_I}$.\n\\begin{conj} Let $I \\subseteq C^\\infty(X)$ be a projective divisor ideal. Then $I$ is divisible by $I_{TX_I}$.\n\\end{conj}\nNote that in general, given a Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and a divisor ideal $I$, there are induced maps $\\varphi\\colon \\Gamma(\\mathcal{A})_I \\to \\Gamma(\\mathcal{A}')_I$ and $\\varphi\\colon I \\cdot \\Gamma(\\mathcal{A}) \\to I \\cdot \\Gamma(\\mathcal{A}')$. In particular this holds upon taking $\\varphi = \\rho_{A}$ and $\\mathcal{A}' = TX$. The following is based on \\cite[Remark 135]{Frejlich11}.\n\\begin{prop}\\label{prop:transLAprojective} Let $\\mathcal{A} \\to X$ be a transitive Lie algebroid and $I$ a projective divisor ideal. Then $\\Gamma(\\mathcal{A})_I$ is a projective module, and the associated bundle $\\mathcal{A}_I$ is a $TX_I$-Lie algebroid. Moreover, $\\mathcal{A}_I$ is isomorphic to the fiber product $\\mathcal{A} \\times_{TX} TX_I$.\n\\end{prop}\n\\bp As $\\mathcal{A}$ is transitive, there is a short exact sequence $0 \\to K \\to \\mathcal{A} \\to TX \\to 0$ where $K = \\ker \\rho_{A}$ is totally intransitive. We can always split this sequence using an Ehresmann connection giving an isomorphism $\\Gamma(\\mathcal{A}) \\cong \\Gamma(TX) \\oplus \\Gamma(K)$. Using this splitting it follows that $\\Gamma(\\mathcal{A})_I \\cong \\Gamma(TX)_I \\oplus \\Gamma(K)$, showing projectivity of $\\Gamma(\\mathcal{A})_I$ and implying that there is an isomorphism $\\mathcal{A}_I \\cong \\mathcal{A} \\times_{TX} TX_I$, from which the fact that $\\mathcal{A}_I$ is a $TX_I$-Lie algebroid follows.\n\\end{proof}\nGiven a surjective Lie algebroid morphism $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and divisor ideal $I$, if the module $\\Gamma(\\mathcal{A}')_I$ is projective, the proof of \\autoref{prop:transLAprojective} shows that $\\Gamma(\\mathcal{A})_I$ is also projective, and that $\\mathcal{A}_I \\cong \\mathcal{A} \\times_{\\mathcal{A}'} \\mathcal{A}'_I$. Sadly the transitivity of \\autoref{prop:transLAprojective} is never satisfied for Lie algebroids of nontrivial divisor-type, so that this inheritance property does not apply to them.\nWe next study to what extent $\\Gamma(\\mathcal{A})_I$ depends on the ideal $I$. Recall that the \\emph{radical} of $I$ is defined as $\\sqrt{I} := \\{f \\in C^\\infty(X) \\, | \\, f^n \\in I \\text{ for some } n \\in \\mathbb{N}\\}$. Then $I \\subseteq \\sqrt{I}$ and $Z_I = Z_{\\sqrt{I}}$.\n\\begin{prop}\\label{prop:locprincidealmodule} Let $I \\subseteq C^\\infty(X)$ be a locally principal ideal. Then $\\Gamma(\\mathcal{A})_I = \\Gamma(\\mathcal{A})_{\\sqrt{I}}$.\n\\end{prop}\n\\bp This can be verified locally. Let $f \\in \\sqrt{I}$ be such that $\\sqrt{I} = \\langle f \\rangle$, so that $I = \\langle f^n \\rangle$ for some $n \\in \\mathbb{N}$. Take $V \\in \\Gamma(\\mathcal{A})_I$, so that $\\mathcal{L}_V f^n = g \\in I$. Then $g = g' f^n$ for some function $g'$, hence $n f^{n-1} \\mathcal{L}_V f = g' f^n$, so that $\\mathcal{L}_V f = \\frac1n g' f$, whence $\\mathcal{L}_V f \\in \\sqrt{I}$. As $f$ generates $\\sqrt{I}$, we conclude that $V \\in \\Gamma(\\mathcal{A})_{\\sqrt{I}}$, and thus $\\Gamma(\\mathcal{A})_I \\subseteq \\Gamma(\\mathcal{A})_{\\sqrt{I}}$. For the other inclusion, take $W \\in \\Gamma(\\mathcal{A})_{\\sqrt{I}}$ and $h \\in I$. Then $h = h' f^n$ for some function $h'$, hence\n\\begin{equation*}\n\t\\mathcal{L}_W h = \\mathcal{L}_W h' f^n = n h' f^{n-1} \\mathcal{L}_W f + f^n \\mathcal{L}_W h'.\n\\end{equation*}\nAs $\\mathcal{L}_W f \\in \\sqrt{I}$, write $\\mathcal{L}_W f = f' f$ for some function $f'$. Then $\\mathcal{L}_W h = (n h' f' + \\mathcal{L}_W h') f^n \\in I$, so that $W \\in \\Gamma(\\mathcal{A})_I$. Thus $\\Gamma(\\mathcal{A})_{\\sqrt{I}} \\subseteq \\Gamma(\\mathcal{A})_I$, and equality follows.\n\\end{proof}\nAs a consequence of \\autoref{prop:locprincidealmodule}, the primary ideal Lie algebroids $\\mathcal{A}_I$ associated to divisor ideals $I$ are only sensitive to $I$ up to taking powers; this is not true for the secondary ideal Lie algebroids $I \\cdot \\mathcal{A}$.\n\\begin{rem}\\label{rem:nonstandard} There are many divisor ideals which are not standard. For example, using \\autoref{prop:locprincidealmodule} we obtain that $\\Gamma(\\mathcal{A})_{I^k} = \\Gamma(\\mathcal{A})_I$ for all $k > 1$. Consequently, if $I$ is a projective divisor ideal, we have that $TX_{I^k} = TX_I$. Hence, if $I$ is moreover standard, we get that $I_{TX_{I^k}} = I_{TX_I} = I$, which is not equal to $I^k$ unless $I$ is trivial, i.e.\\ $I^k$ is not standard.\n\\end{rem}\n\\begin{rem}\\label{rem:preservingradical} Analogously to \\autoref{rem:preservingideals}, a slight adaptation of the proof of \\autoref{prop:locprincidealmodule} shows that more generally, given a collection of vector fields $\\mathcal{F} \\subseteq \\Gamma(\\mathcal{A})$ preserving an ideal $I$, i.e.\\ such that $\\mathcal{L}_v I \\subseteq I$ for all $v \\in \\mathcal{F}$, the collection $\\mathcal{F}$ also preserves $\\sqrt{I}$. This is a purely algebraic statement and can be found in \\cite[Lemma 3.3.2]{Dixmier96}.\n\\end{rem}\nAs $Z_I$ is the degeneracy locus of both $I\\cdot \\mathcal{A}$ and $\\mathcal{A}_I$, \\autoref{prop:degenlocus} gives the following.\n\\begin{prop}\\label{prop:ziaiinvariant} Let $I \\subseteq C^\\infty(X)$ be a divisor ideal such that $\\mathcal{A}_I$ exists. Then $Z_I$ is $I \\cdot \\mathcal{A}$- and $\\mathcal{A}_I$-invariant if it is smooth, i.e.\\ $I \\cdot \\Gamma(\\mathcal{A})$ and $\\Gamma(\\mathcal{A}_I)$ have anchor images tangent to $Z_I$.\n\\end{prop}\nLet $I$ and $I'$ be two divisor ideals. If $\\mathcal{A}_{I \\cdot I'}$ exists, then so do $\\mathcal{A}_I$ and $\\mathcal{A}_{I'}$, and we see that $\\mathcal{A}_{I \\cdot I'} = \\mathcal{A}_I \\times_{TX} \\mathcal{A}_{I'}$ is their fiber product. Moreover, $\\mathcal{A}_{I \\cdot I'}$ is both an $\\mathcal{A}_I$- and an $\\mathcal{A}_{I'}$-Lie algebroid (again, if it exists). In favorable cases, the $\\mathcal{A}$-divisor of $\\mathcal{A}_{I \\cdot I'}$ is the product of divisors of $\\mathcal{A}_I$ and $\\mathcal{A}_{I'}$. For example this holds if $\\mathcal{A} = TX$ and $I \\cdot I'$ is standard.\n\\begin{rem} While $\\Gamma(\\mathcal{A})_{I \\cdot I'}$ being projective implies both $\\Gamma(\\mathcal{A})_I$ and $\\Gamma(\\mathcal{A})_{I'}$ are, the converse is not immediate if $Z_I$ and $Z_{I'}$ are not disjoint. For example, one can take $\\mathcal{A} = TX$, $I = I_Z$ and $I' = I_{Z'}$ for hypersurfaces $Z,Z' \\subseteq X$ which have nontransverse intersection.\n\\end{rem}\n\\subsubsection{Examples}\n\\label{exa:ideallalgebroids}\nIn this section we discuss some examples of ideal Lie algebroids constructed using the divisor examples of \\autoref{sec:divexamples}. In general it is quite difficult to determine when a divisor ideal is projective, comparable to finding free divisors in algebraic geometry.\n\\begin{exa}[Trivial divisors] Let $I = C^\\infty(X)$ be the trivial divisor ideal. Then\n\\begin{equation*}\n\tI \\cdot \\Gamma(TX) = \\Gamma(TX)_I = \\Gamma(TX),\n\\end{equation*}\nso that the ideal Lie algebroids satisfy $I \\cdot TX = TX_I = TX$. The same holds for any Lie algebroid $\\mathcal{A}$, in that $I \\cdot \\mathcal{A} = \\mathcal{A}_I = \\mathcal{A}$. In particular $I$ is both projective and standard. \n\\end{exa}\n\\begin{exa}[Log divisors]\\label{exa:zvx} Let $Z \\subseteq X$ be a closed submanifold with vanishing ideal $I_Z$. Then $\\Gamma(TX)_{I_Z}$ consists of all vector fields tangent to $Z$. It is projective if and only if $Z$ has codimension one. In this case $I_Z$ is a log divisor ideal and $\\mathcal{A}_Z := TX_{I_Z}$ is the \\emph{log-tangent bundle}\\footnote{It is also called the \\emph{$b$-tangent bundle} ${}^b TX$, introduced by Melrose when $Z = \\partial X$ (see \\cite{Melrose93} and \\cite{NestTsygan96, GuilleminMirandaPires14}).} $TX(-\\log Z)$. Further, $I_Z \\cdot \\Gamma(TX)$ consists of all vector fields which vanish on $Z$. As with $\\Gamma(TX)_{I_Z}$, it is projective if and only if $Z$ has codimension one, in which case $\\mathcal{B}_Z := I_Z \\cdot TX$ is the \\emph{zero tangent bundle} (also denoted by ${}^0 TX$). Log divisors are standard (\\cite[Proposition 3.4.6]{Klaasse17}), as locally we have\n\\begin{equation*}\n\t\\Gamma(\\mathcal{A}_Z) = \\Gamma(TX)_{I_Z} = \\langle z \\partial_z, \\partial_{x_i} \\rangle \\quad \\text{and} \\quad \\Gamma(\\mathcal{B}_Z) = I_Z \\cdot \\Gamma(TX) = \\langle z \\partial_z, z \\partial_{x_i} \\rangle.\n\\end{equation*}\n\\end{exa}\n\\autoref{exa:zvx} extends to normal-crossing and star log divisors $I_{\\underline{Z}}$. In both cases, there is a projective module $\\Gamma(TX)_{I_{\\underline{Z}}}$ defining the log-tangent bundle $TX(-\\log \\underline{Z})$, which locally is generated by $\\langle z_j \\partial_{z_j}, \\partial_{x_i} \\rangle$ when $\\underline{Z} = \\bigcup_j Z_j$ with $Z_j = \\{z_j = 0 \\}$ (see \\cite{GualtieriLiPelayoRatiu17} and \\cite{Lanius17} respectively). In the former case, the log-tangent bundle $TX(-\\log \\underline{Z})$ is the fiber product of the individual log-tangent bundles $TX(-\\log Z_j)$ associated to the log divisors $I_{Z_j}$ (c.f.\\ below \\autoref{prop:ziaiinvariant}).\n\\begin{exa}[Elliptic divisors]\\label{exa:ellvx} Let $|D| = (R,q)$ be an elliptic divisor with elliptic divisor ideal $I_{|D|}$. Then the module $\\Gamma(TX)_{I_{|D|}}$ is projective, defining the \\emph{elliptic tangent bundle} $\\mathcal{A}_{|D|} = TX(-\\log |D|)$ \\cite{CavalcantiGualtieri18}, which is locally given by $\\Gamma(\\mathcal{A}_{|D|}) = \\Gamma(TX)_{I_{|D|}} = \\langle r \\partial_r, \\partial_{\\theta}, \\partial_{x_i} \\rangle$ in normal polar coordinates $(r,\\theta)$ to $D$ in which $I_{|D|} = \\langle r^2 \\rangle$. From the fact that $r \\partial_r \\wedge \\partial_{\\theta} = r^2 \\partial_x \\wedge \\partial_y$ where $r^2 = x^2 + y^2$ we see that elliptic divisors are standard (see \\cite[Proposition 3.4.17]{Klaasse17}).\n\\end{exa}\n\\begin{rem} Morphisms of divisors often give rise to Lie algebroid morphisms between their primary ideal Lie algebroids. Indeed, in \\cite{Klaasse17,CavalcantiKlaasse18} it is shown that a morphism between log divisor ideals leads to an induced Lie algebroid morphism between their resulting log-tangent bundles. Similarly for morphisms between elliptic divisor ideals. Morphisms between elliptic and log divisors are considered in \\cite{CavalcantiKlaasse18} as boundary maps, and lead to Lie algebroid morphism from the elliptic tangent bundle to the log-tangent bundle. Finally, regarding morphisms between log and elliptic divisors one has to be more careful, see \\cite{KlaasseLi18}.\n\\end{rem}\n\\begin{exa}[Complex log divisors]\\label{exa:complexlogtgnt} Let $D = (U,\\sigma)$ be a complex log divisor with ideal $I_\\sigma$. Then $\\Gamma(TX_\\mathbb{C})_{I_D}$ is a projective module of complex vector fields, defining the \\emph{complex log-tangent bundle} $TX_{\\mathbb{C}}(- \\log D)$, locally given by $\\langle w \\partial_w, \\partial_{\\overline{w}}, \\partial_{x_i} \\rangle$ in coordinates where $I_\\sigma = \\langle w \\rangle$ (see \\cite{CavalcantiGualtieri18}). Because $w \\partial_w \\wedge \\partial_{\\overline{w}} = - 2i w \\partial_x \\wedge \\partial_y$, we see that complex log divisors are standard. This complex Lie algebroid is also denoted by $\\mathcal{A}_D \\to X$, and we will not use it in this paper.\n\\end{exa}\nWe discuss one further example, obtained from an elliptic-log divisor. Let $W = Z \\otimes |D|$ be an elliptic-log divisor with divisor ideal $I_W$. Then the primary ideal Lie algebroid $\\mathcal{A}_W := \\mathcal{A}_{I_W}$, the \\emph{elliptic-log tangent bundle} $TX(-\\log W)$, exists and is both an $\\mathcal{A}_Z$ and an $\\mathcal{A}_{|D|}$-Lie algebroid, as it is the fiber product of the two. The following was pointed out to us by Gil Cavalcanti.\n\\begin{prop}\\label{prop:elllogalgebroid} Let $(X,Z)$ be a log pair and $|D| \\subseteq Z$ an elliptic divisor. Then the module $\\Gamma(TX)_{I_W}$ is projective, so that $\\mathcal{A}_{W}$ is a Lie algebroid. Its divisor ideal is given by $I_{\\mathcal{A}_W} = I_W$.\n\\end{prop}\n\\bp By definition $\\Gamma(TX)_{I_W}$ consists of vector fields preserving $I_W = I_Z \\cdot I_{|D|}$. It is immediate (see \\cite{Klaasse17}) that all vector fields tangent to $D$ belong to $\\Gamma(TX)_{I_W}$. Hence the associated sheaf is locally free if around points in $D$ it contains only two independent vector fields normal to $D$, as $D$ has codimension two. Away from $D$ we have $I_W = I_Z$, so that there $\\Gamma(TX)_{I_W} = \\Gamma(TX)_{I_Z}$ which we know to be locally free. Let $(x,y)$ be normal coordinates around $p \\in D$ such that $I_W = \\langle x (x^2 + y^2) \\rangle$. Set $w(x,y) := x (x^2 + y^2)$ and take $V = a \\partial_x + b \\partial_y$. Note that $V$ belongs to $\\Gamma(TX)_{I_W}$ if and only if the function $f = \\mathcal{L}_V (\\log w)$ is smooth. We compute that\n\\begin{equation*}\nf = \\mathcal{L}_V(\\log w) = \\mathcal{L}_V(\\log x) + \\mathcal{L}_V(\\log(x^2 + y^2)) = a x^{-1} + (2a x + 2 b y)(x^2+y^2)^{-1},\n\\end{equation*}\nso by multiplying by $x^2 + y^2$ and rearranging we obtain $(x^2 + y^2) f - 2a x - 2b y = a x^{-1} (x^2 + y^2)$. The left-hand side is smooth, hence so must be the right-hand side, i.e.\\ $a$ must be divisible by $x$. Hence $a = x \\alpha$ for some smooth $\\alpha$. This implies that $f = \\alpha + (2 x^2 \\alpha + 2 b y)(x^2 + y^2)^{-1}$, so that $\\lambda_1 := (x^2 \\alpha + b y)(x^2 + y^2)^{-1}$ must be smooth. Rewriting this we obtain that\n\\begin{equation*}\n\tb = \\lambda_1 y + (\\lambda_1 - \\alpha)x^2 y^{-1},\n\\end{equation*}\nand as $b$ is smooth, $y$ divides $\\lambda_1 - \\alpha$. Set $\\lambda_2 := (\\lambda_1 - \\alpha) y^{-1}$. Given smooth functions $\\lambda_1$ and $\\lambda_2$ we can now express $a$ and $b$ as $a = x (\\lambda_1 - y \\lambda_2)$ and $b = \\lambda_1 y + \\lambda_2 x^2$. This implies that $V$ is given by $V = \\lambda_1 (x \\partial_x + y \\partial_y) + \\lambda_2 (-x y \\partial_x + x^2 \\partial_y)$. This shows that $V$ must lie in the two-dimensional span $\\langle x \\partial_x + y \\partial_y, x(y \\partial_x - x \\partial_y) \\rangle$. To finish the proof of local freeness, we check that these generators preserve $I_W$. This follows, as we readily compute:\n\\begin{itemize}\n\t\\item \\makebox[3cm][l]{$\\mathcal{L}_{x \\partial_x + y \\partial_y}(\\log w)$} $= x \\partial_x \\log x + (x \\partial_x + y \\partial_y) \\log(x^2 + y^2) = 1 + 2 = 3$,\n\t\\item \\makebox[3cm][l]{$\\mathcal{L}_{x(y \\partial_x - x \\partial_y)}(\\log w)$} $= x y \\partial_x \\log x + x y \\partial_x \\log(x^2 + y^2) - x^2 \\partial_y \\log(x^2 + y^2)$\n\t\\item[] \\makebox[3cm][l]{} $= y + (2 x^2 y - 2 x^2 y)(x^2 + y^2)^{-1} = y$.\n\\end{itemize}\nWe conclude that $\\mathcal{A}_W$ exists and that locally we have (with slight abuse of notation)\n\\begin{equation*}\n\\Gamma(\\mathcal{A}_W) = \\Gamma(TX)_{I_W} = \\langle x \\partial_x + y \\partial_y, x(y \\partial_x - x \\partial_y) \\rangle \\oplus \\Gamma(TD).\n\\end{equation*}\nThe final statement of this proposition follows from the realization that\n\\begin{equation*}\n(x \\partial_x + y \\partial_y) \\wedge x(y \\partial_x - x \\partial_y) = x (x^2 + y^2) \\partial_x \\wedge \\partial_y = w \\partial_x \\wedge \\partial_y. \\qedhere\n\\end{equation*}\n\\end{proof}\nWe can alternatively write the generators of $\\mathcal{A}_W$ using a polar coordinate system $(r,\\theta)$ normal to $D$ supplied by $I_{W} = \\langle r^3 \\cos \\theta \\rangle$. In such a coordinate system, we have that\n\\begin{equation*}\n\\Gamma(\\mathcal{A}_W) = \\langle r \\partial_r, r \\cos \\theta \\partial_\\theta \\rangle + \\Gamma(TD).\n\\end{equation*}\nIn particular, \\autoref{prop:elllogalgebroid} says that $I_W$ is standard. The elliptic-log tangent bundle is further explored in \\autoref{exa:ellipticlogresidue} and \\cite{KlaasseLanius18}, the latter computing its Lie algebroid cohomology.\n\nA slightly different example is provided by the $b^k$-tangent bundles \\cite{Scott16}. The idea here is that instead of considering derivations preserving a certain divisor ideal $I$ (which would give its primary ideal Lie algebroid), we consider those which send $I$ to $I^k$ in a certain sense.\n\nLet $(X,Z)$ be a log pair with $\\iota_Z\\colon Z \\hookrightarrow X$ the inclusion, $k \\geq 1$ be a given integer. The \\emph{sheaf of $k$-jets} at $Z$ is $\\mathcal{J}_Z^k := \\iota_Z^{-1}(C^\\infty(X) \/ I_Z^{k+1})$. A \\emph{$k$-jet} at $Z$ is a global section of $\\mathcal{J}_Z^k$. Denote the set of $k$-jets at $Z$ by $J_Z^k$.\nGiven a function $f$ defined in a neighbourhood of $Z$, let $[f]^k_Z \\in J_Z^k$ denote the $k$-jet at $Z$ represented by $f$. For a given $k$-jet at $Z$, $j \\in J_Z^k$, write $f \\in j$ if $[f]_Z^k = j$. \n\\begin{exa}[$b^k$-tangent bundles, \\cite{Scott16}]\\label{exa:bktangentbundle} Let $j_{k-1} \\in J_Z^{k-1}$ be a choice of $(k-1)$-jet and define the sheaf of vector fields $\\Gamma(TX)_{I^k_Z, j_{k-1}} := \\{v \\in \\Gamma(TX) \\, | \\, \\mathcal{L}_v f \\in I_Z^k \\text{ for all } f \\in j_{k-1}\\}$. One verifies that this is a locally free sheaf of involutive submodules of $\\Gamma(TX)$, and that given $v \\in \\mathcal{V}_X(I_Z)$ and $f \\in C^\\infty(X)$, the jet $[\\mathcal{L}_v f]_Z^{k-1}$ depends only on $[f]_Z^{k-1}$. This defines the \\emph{$b^k$-tangent bundle} $\\mathcal{A}_Z^k \\to X$ by $\\Gamma(\\mathcal{A}_Z^k) \\cong \\Gamma(TX)_{I^k_Z, j_{k-1}}$, which is locally given by $\\Gamma(\\mathcal{A}_Z^k) = \\langle z^k \\partial_z, \\partial_{x_i} \\rangle$ for $z \\in j$.\n\\end{exa}\nNote that when $k = 1$ the jet data is vacuous, so that $\\mathcal{A}_Z^1 = \\mathcal{A}_Z$, the log-tangent bundle. Further discussion of this process can be found in \\cite{Scott16,Klaasse17}. The same construction described above can also be performed with smooth divisor ideals other than $I_Z$, where one uses a choice of $k$-jet at $Z_I$, i.e.\\ a global section of $\\mathcal{J}^k_I := \\iota_{Z_I}^{-1}(C^\\infty(X) \/ I^{k+1})$, for $\\iota_{Z_I}\\colon Z_I \\hookrightarrow X$ the inclusion. The sheaf of vector fields then becomes $\\Gamma(TX)_{I^k, j_{k-1}} := \\{v \\in \\Gamma(TX) \\, | \\, \\mathcal{L}_v f \\in I^k \\text{ for all } f \\in j_{k-1}\\}$.\n\\begin{rem} Let $I \\subseteq C^\\infty(X)$ be a projective smooth divisor ideal. Then its associated primary ideal Lie algebroid $TX_I$ can be restricted to $Z_I$, because it is the degeneracy locus of $TX_I$. Assuming that $Z_I$ is a transitive invariant submanifold for $TX_I$, the restriction $TX_I|_{Z_I}$ is then a Lie subalgebroid of ${\\rm At}(NZ_I) \\to Z_I$, the Atiyah algebroid of the normal bundle. The latter is a transitive Lie algebroid with rank equal to ${\\rm rank}({\\rm At}(NZ_I)) = \\dim(Z_I) + {\\rm codim}(Z_I)^2$. If $Z_I$ is not transitive, then instead there is only an almost-injective Lie algebroid morphism from $TX_I|_{Z_I}$ to ${\\rm At}(NZ_I)$. As ${\\rm rank}(TX_I|_{Z_I}) = {\\rm rank}(TX_I) = \\dim(X)$, this means that in order for $I$ to be projective, the vector fields normal to $Z_I$ which preserve $I$ must cut down the isotropy of ${\\rm At}(NZ_I)$ in dimension from ${\\rm codim}(Z_I)^2$ to ${\\rm codim}(Z_I)$. Related to this is the discussion in \\autoref{sec:divexamples} on Morse--Bott divisor ideals being locally homogeneous around $Z_I$.\n\t\n\tFor example, if $I = I_Z$ is a log divisor ideal, then the above implies that $TX_{I_Z}|_Z \\cong {\\rm At}(NZ)$. For elliptic divisor ideals, instead $TX_{I_{|D|}}|_D$ is a corank-two Lie subalgebroid of ${\\rm At}(ND)$.\n\\end{rem}\n\\subsection{Elementary modifications}\n\\label{sec:elementarymodifications}\nIn this section we discuss a process called \\emph{elementary modification} \\cite{GualtieriLi14, Li13} (partially also \\emph{rescaling} \\cite{Lanius16, Melrose93}), by which, given a Lie algebroid, new Lie algebroids are constructed by specifying a projective singular Lie subalgebroid using extra data. In loc.\\ cit.\\ this is done using Lie algebroids supported on hypersurfaces, but we consider a generalization of this procedure by using Lie algebroids supported on the support of a smooth divisor ideal. When using a log divisor ideal (see \\autoref{exa:log}), one recovers the procedures of loc.\\ cit. There are two procedures which are in a sense dual to another, referred to as lower and upper elementary modification respectively. As noted in \\cite{Li13}, these processes are known in algebraic geometry under the name of Hecke modifications. Recall from \\autoref{defn:smoothdivisor} that a divisor ideal $I \\subseteq C^\\infty(X)$ is smooth if its support $Z_I \\subseteq X$ is a smooth submanifold.\n\\subsubsection{Upper elementary modification}\nWe first describe the process of upper modification (see \\cite{Li13} for the case when $I = I_Z$), which uses surjective comorphisms. Let $I$ be a smooth divisor ideal with ideal sheaf $\\mathcal{I}$, and let $\\mathcal{A} \\to X$ and $\\mathcal{B} \\to Z_I$ be vector bundles. Assume the existence of a surjective bundle comorphism $(\\varphi;i_{Z_I})\\colon (\\mathcal{B},Z_I) \\dashedrightarrow (\\mathcal{A},X)$, i.e.\\ the map $\\varphi^*\\colon i_{Z_I}^* \\mathcal{A} \\to \\mathcal{B}$ is surjective, and has kernel $\\mathcal{K} := \\ker \\varphi \\to Z_I$. From this data, we define a $C^\\infty(X)$-module\n\\begin{equation*}\n\\Gamma(\\mathcal{A};(\\mathcal{B},I)) := \\{v \\in \\Gamma(\\mathcal{A}) \\otimes \\mathcal{I}^{-1} \\, | \\, \\mathcal{I} \\cdot v \\in \\Gamma(\\mathcal{A},\\mathcal{K})\\},\n\\end{equation*}\nwhere it is immediate that $\\Gamma(\\mathcal{A}) \\subseteq \\Gamma(\\mathcal{A};(\\mathcal{B},I))$. Here $\\mathcal{I}^{-1}$ is the inverse of $\\mathcal{I}$ as a sheaf, and we make use of the canonical isomorphism $\\mathcal{I} \\otimes \\mathcal{I}^{-1} \\cong \\mathcal{C}^\\infty_X$. Alternatively, we can choose local generators for the sheaf $\\mathcal{I}$ and its inverse and check that the definition is independent of such a choice, or first realize $\\mathcal{I}$ by a divisor using \\autoref{prop:locprincideal}. It is immediate that $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ is locally finitely generated because $I$ and $\\Gamma(\\mathcal{A})$ are. The main point of this construction is:\n\\begin{lem}\\label{lem:uppermodprojective} Under the above assumptions, the module $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ is projective.\n\\end{lem}\n\\bp Trivialize $\\mathcal{B}^m \\to Z_I$ around a point in $Z_I$ with basis of sections $(w_1,\\dots,w_m)$, and let $f \\in \\mathcal{I}$ be a local generator. Choose a complementary basis of sections $(v_{m+1},\\dots,v_n)$ for $\\mathcal{K}$, so that $(w_1,\\dots,w_m,v_{m+1},\\dots,v_n)$ is a local basis of $\\Gamma(\\mathcal{A}^n)$. Then $(w_1,\\dots,w_m, f^{-1} v_{m+1},\\dots, f^{-1} v_n)$ is a local basis of $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$, showing projectivity, as $\\Gamma(\\mathcal{A};(\\mathcal{B},I)) \\cong \\Gamma(\\mathcal{A})$ away from $Z_I$.\n\\end{proof}\nBy the Serre--Swan correspondence there exists a vector bundle\n\\begin{equation*}\n\t\\{\\mathcal{A}\\:(\\mathcal{B},I)\\} \\to X \\quad \\text{defined by} \\quad \\Gamma(\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}) \\cong \\Gamma(\\mathcal{A};(\\mathcal{B},I)).\n\\end{equation*}\nNext, assume that $\\mathcal{A} \\to X$ and $\\mathcal{B} \\to Z_I$ are anchored and $\\varphi$ is an anchored comorphism, i.e.\\ $i_{Z_I}^* (\\mathcal{L}_{\\rho_{A}(v)} f) = \\mathcal{L}_{\\rho_\\mathcal{B}(\\varphi(v))}(i_{Z_I}^*(f))$ for all $v \\in \\Gamma(\\mathcal{A})$ and $f \\in C^\\infty(X)$. Alternatively, the anchor maps should satisfy $\\rho_{\\mathcal{B}}(\\varphi^*(v)) \\sim_\\varphi \\rho_{\\mathcal{A}}(v)$ for all $v \\in \\Gamma(\\mathcal{A})$.\n\\begin{lem}[c.f.\\ \\cite{Li13}]\\label{lem:uppermodanchored} Under the above assumptions, the bundle $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ is anchored.\n\\end{lem}\n\\bp The anchored comorphism condition implies $\\mathcal{K} \\subseteq \\ker(\\rho_{A}|_{Z_I})$, so that for $v \\in \\Gamma(\\mathcal{A},K)$ we have $\\rho_{A}(v) \\in \\Gamma(TX,TZ_I)$, which shows that $\\rho_{A}$ lifts to an anchor for $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$.\n\\end{proof}\nThere is an induced (anchored) morphism $\\mathcal{A} \\to \\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ with isomorphism locus $Z_I$. Finally, assume that $\\mathcal{A} \\to X$ and $\\mathcal{B} \\to Z_I$ are Lie algebroids and $\\varphi$ is a surjective Lie algebroid comorphism, i.e.\\ $\\varphi$ further satisfies $\\varphi^*([v,w]) = [\\varphi^*(v),\\varphi^*(w)]$ for all $v,w \\in \\Gamma(\\mathcal{A})$.\n\\begin{lem}\\label{lem:uppermodinvolutive} Under the above assumptions, the module $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ has an induced bracket.\n\\end{lem}\n\\bp Because $\\varphi^*$ preserves brackets, there is an induced Lie bracket on $\\Gamma(\\mathcal{A};(\\mathcal{B},I))$ inherited from the module $\\Gamma(\\mathcal{A})$ through the natural map $\\Gamma(\\mathcal{A}) \\to \\Gamma(\\mathcal{A};(\\mathcal{B},I))$.\n\\end{proof}\n\\autoref{lem:uppermodprojective}, \\autoref{lem:uppermodanchored} and \\autoref{lem:uppermodinvolutive} combine to the following definition (c.f.\\ \\cite{Li13}).\n\\begin{defn}\\label{defn:uppermod} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I \\subseteq C^\\infty(X)$ a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ a Lie algebroid with a surjective Lie algebroid comorphism onto $\\mathcal{A}$. The \\emph{$(\\mathcal{B},I)$-upper modification} of $\\mathcal{A}$ is the \n\n\tLie algebroid $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ for which $\\Gamma(\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}) \\cong \\Gamma(\\mathcal{A};(\\mathcal{B},I))$.\n\\end{defn}\nIn this case there is a Lie algebroid morphism $\\mathcal{A} \\to \\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ making $\\mathcal{A}$ into an $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$-Lie algebroid, and the $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$-divisor ideal of $\\mathcal{A}$ is given by $I^k$, where $k = \\dim(\\mathcal{B})$.\n\\begin{rem} From the divisor ideal of the induced morphism $\\mathcal{A} \\to \\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ we see that it is only possible for $\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}$ to be anchored if the divisor of $\\mathcal{A}$ is divisible by $I^k$. This places an often nontrivial restriction on when upper modifications can be Lie algebroids.\n\\end{rem}\nThere is a commutativity property for upper modifications, which is readily verified.\n\\begin{prop} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I,I' \\subseteq C^\\infty(X)$ smooth divisor ideals and $(\\mathcal{B},Z_I)$, $(\\mathcal{B}',Z_{I'})$ Lie algebroids equipped with surjective Lie algebroid comorphisms onto $\\mathcal{A}$. Then upper modification at $Z_I$ and $Z_{I'}$ commute, i.e. we have isomorphisms\n\t%\n\t\\begin{equation*}\n\t\\{\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}\\:(\\mathcal{B}',I')\\} \\cong \\{\\{\\mathcal{A}\\:(\\mathcal{B}',I')\\}\\:(\\mathcal{B},I)\\}.\n\t\\end{equation*}\n\t%\n\\end{prop}\n\\subsubsection{Lower modification}\nWe turn to the dual procedure to upper modification, namely lower modification (or rescaling). This uses injective morphisms instead of surjective comorphisms.\n \nLet $I$ be a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ be a vector subbundle of a vector bundle $\\mathcal{A} \\to X$. To define an appropriate $C^\\infty(X)$-module in this setting, we remark that injective morphisms dualize to surjective comorphisms. That is, the bundle inclusion $(\\varphi,i_{Z_I})\\colon (\\mathcal{B},Z_I) \\to (\\mathcal{A},X)$ dualizes to the surjection $(\\varphi;i_{Z_I})\\colon (\\mathcal{B}^*,Z_I) \\dashrightarrow (\\mathcal{A}^*,X)$. We can thus perform upper modification to obtain the vector bundle\n\\begin{equation*}\n\t\\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\} \\to X \\quad \\text{with} \\quad \\Gamma(\\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}) \\cong \\Gamma(\\mathcal{A}^*\\:(\\mathcal{B}^*,I)).\n\\end{equation*}\nThis comes equipped with a natural bundle morphism $\\mathcal{A}^* \\to \\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}$. Dualizing again, using $(\\mathcal{A}^*)^* \\cong \\mathcal{A}$ we obtain a comorphism $\\mathcal{A} \\dashrightarrow \\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}^*$. Reinterpreted, this says that there is a bundle morphism $\\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}^* \\to \\mathcal{A}$. This motivates the following definitions:\n\\begin{equation*}\n\t[\\mathcal{A}\\:(\\mathcal{B},I)] := \\{\\mathcal{A}^*\\:(\\mathcal{B}^*,I)\\}^* \\quad \\text{and} \\quad \\Gamma(\\mathcal{A},(\\mathcal{B},I)) := \\Gamma([\\mathcal{A}\\:(\\mathcal{B},I)]).\n\\end{equation*}\n\\begin{rem} We had to proceed by dualizing twice because of the use of a smooth divisor ideal $I$. When $I = I_Z$, one can merely define $\\Gamma(\\mathcal{A},(\\mathcal{B},I_Z)) := \\Gamma(\\mathcal{A},\\mathcal{B})$ as in \\cite{Melrose93,Lanius16,GualtieriLi14}. Note here that the `dual definition' $\\{v \\in \\Gamma(\\mathcal{A}) \\otimes \\mathcal{I} \\, | \\, \\mathcal{I}^{-1} \\cdot v \\in \\Gamma(\\mathcal{A},\\mathcal{B})\\}$ does not make sense.\n\\end{rem}\nNow that we have the module of interest, it can be described locally, as in \\autoref{lem:uppermodprojective}. Trivialize $\\mathcal{B}^m \\to Z_I$ around a point in $Z_I$ with basis of sections $(w_1, \\dots w_m)$, and let $f \\in \\mathcal{I}$ be a local generator. Extend this basis to a local basis $(w_1,\\dots,w_m,v_{m+1},\\dots,v_n)$ of $\\Gamma(\\mathcal{A}^n)$. Then $(w_1,\\dots,w_m, f v_{m+1}, \\dots, f v_n)$ is a local basis for $\\Gamma(\\mathcal{A},(\\mathcal{B},I))$. This also shows projectivity.\n\nWhen $\\mathcal{A} \\to X$ is an anchored bundle and $\\mathcal{B} \\to Z_I$ an anchored subbundle of $\\mathcal{A}$, the bundle $[\\mathcal{A}\\:(\\mathcal{B},I)]$ becomes anchored by inheriting an anchor from $\\rho_{A}\\colon \\mathcal{A} \\to TX$. Note that there is an induced (anchored) morphism $[\\mathcal{A}\\:(\\mathcal{B},I)] \\to \\mathcal{A}$ with isomorphism locus $Z_I$. Note that $[\\mathcal{A}\\:(\\mathcal{B},I)]$ is \\emph{not} an anchored subbundle of $\\mathcal{A}$ (except in the trivial case when $I = C^\\infty(X)$).\n\nNext assume that $\\mathcal{A} \\to X$ is a Lie algebroid and $\\mathcal{B} \\to Z_I$ is a Lie subalgebroid.\n\\begin{lem}\\label{lem:lowermodinvolutive} Under the above assumptions, the submodule $\\Gamma(\\mathcal{A},(\\mathcal{B},I)) \\subseteq \\Gamma(\\mathcal{A})$ is involutive.\n\\end{lem}\n\\bp The submodule $\\Gamma(\\mathcal{A},\\mathcal{B}) \\subseteq \\Gamma(\\mathcal{A})$ is involutive by \\autoref{prop:liesubalgeroid}. To extend this property to $\\Gamma(\\mathcal{A},(\\mathcal{B},I))$ we can make use of the local description above. Let $f \\in \\mathcal{I}$ be a local generator and consider two sections $f s, f s' \\in \\Gamma(\\mathcal{A},(\\mathcal{B},I))$ with $s \\in \\Gamma(\\mathcal{A})$ but not in $\\Gamma(\\mathcal{A},\\mathcal{B})$ (here $g \\in \\mathcal{I}$ can be written as $g = g' f$, but then $g'$ can be absorbed into $s$). Then we have:\n\\begin{equation*}\n\t[f s, f s']_\\mathcal{A} = f^2 [s,s']_\\mathcal{A} + f \\left((\\mathcal{L}_{\\rho_{A}(s)} f) s' - (\\mathcal{L}_{\\rho_{A}(s')} f) s\\right).\n\\end{equation*}\nWe see from this that $[fs,f s']_\\mathcal{A} \\in \\mathcal{I} \\otimes \\Gamma(\\mathcal{A})$, as in \\autoref{lem:liesubalgs}. The other cases are similar.\n\\end{proof}\nThus, making use of \\autoref{lem:lowermodinvolutive} we see that $\\Gamma(\\mathcal{A},(\\mathcal{B},I))$ is a projective singular Lie subalgebroid of $\\mathcal{A}$ in the sense of \\autoref{defn:singularliealgebroid}. This leads to the following.\n\\begin{defn}[{\\cite{GualtieriLi14, Lanius16, Melrose93}}]\\label{defn:lowermod} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I \\subseteq C^\\infty(X)$ a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ a Lie subalgebroid of $\\mathcal{A}$. Then the \\emph{$(\\mathcal{B},I)$-lower modification} of $\\mathcal{A}$ is the $\\mathcal{A}$-Lie algebroid $[\\mathcal{A}\\:(\\mathcal{B},I)] \\to \\mathcal{A}$ for which $\\Gamma([\\mathcal{A}\\:(\\mathcal{B},I)]) \\cong \\Gamma(\\mathcal{A},(\\mathcal{B},I))$.\n\\end{defn}\nIt is immediate that the $\\mathcal{A}$-divisor ideal of $[\\mathcal{A}\\:(\\mathcal{B},I)]$ is given by $I^k$, where $k = {\\rm codim}(\\mathcal{B})$.\n\\begin{rem} When $I = I_Z$, the above procedure is also called \\emph{rescaling} \\cite{Lanius16, Melrose93}; we then denote $[\\mathcal{A}\\:(\\mathcal{B},I_Z)]$ by $[\\mathcal{A}\\:\\mathcal{B}]$ or ${}^{\\mathcal{B}} \\mathcal{A}$ and refer to it as the \\emph{$(\\mathcal{B},Z)$-rescaling} of $\\mathcal{A}$.\n\\end{rem}\n\\begin{rem}\\label{rem:bbprimerescaling} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I$ a smooth divisor ideal and $\\mathcal{B}, \\mathcal{B}' \\to Z_I$ two Lie subalgebroids of $\\mathcal{A}$ such that $\\mathcal{B} \\subseteq \\mathcal{B}'$. Then $[\\mathcal{A}\\:(\\mathcal{B},I)]$ is a $[\\mathcal{A}\\:(\\mathcal{B}',I)]$-Lie algebroid.\n\\end{rem}\nOne can always modify using $0_{Z_I} \\subseteq \\mathcal{A}|_{Z_I}$, the trivial Lie subalgebroid, with\n\\begin{equation*}\n\t[\\mathcal{A}\\:(0_{Z_I},I)] = I \\cdot \\mathcal{A},\n\\end{equation*}\nwhich is the secondary ideal Lie algebroid with section module $I \\cdot \\Gamma(\\mathcal{A})$. Thus lower modification by $0_{Z_I}$ is the harshest, while lower modification using $\\mathcal{B} = \\rho_\\mathcal{A}^{-1}(T(Z_I))$ (if it exists, c.f.\\ \\autoref{rem:inverseimagesmooth}) is the mildest. Note that $0_{Z_I}$-modification is not idempotent.\nThe process of lower modification is commutative when performed relative to disjoint submanifolds.\n\\begin{prop}\\label{prop:lowermodifycommute} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I,I' \\subseteq C^\\infty(X)$ smooth divisor ideals and $(\\mathcal{B}, Z_I)$, $(\\mathcal{B}',Z_{I'})$ Lie subalgebroids of $\\mathcal{A}$ such that $Z_I \\cap Z_{I'} = \\emptyset$. Then lower modification at $Z_I$ and $Z_{I'}$ commute and coincides with taking the fiber product, i.e. we have isomorphisms\n\\begin{equation*}\n\t[[\\mathcal{A}\\:(\\mathcal{B},I)]:(\\mathcal{B}',I')] \\cong [\\mathcal{A}\\:(\\mathcal{B},I)] \\oplus_{\\mathcal{A}} [\\mathcal{A}\\:(\\mathcal{B}',I')] \\cong [[\\mathcal{A}\\:(\\mathcal{B}',I')]:(\\mathcal{B},I)].\n\\end{equation*}\n\\end{prop}\nThe proof of the above is immediate upon noticing we can view $(\\mathcal{B}',Z_{I'})$ as a Lie subalgebroid of $[\\mathcal{A}\\:(\\mathcal{B},I)]$ because the latter is isomorphic to $\\mathcal{A}$ outside $Z_I$, and $Z_I \\cap Z_{I'} = \\emptyset$.\n\\begin{rem} Elementary modification can also be performed in an even slightly more general setting. Namely, instead of a smooth divisor ideal one can use a $C^\\infty(X)$-module $\\mathcal{M}$ that is locally free of rank one, and a suitable vector bundle supported on its smooth support.\n\\end{rem}\nUpper modification is a left inverse to lower modification, and vice versa regarding a right inverse when this makes sense. Often the morphism used is the anchor map on sections.\n\\begin{prop} Let $\\mathcal{A} \\to X$ be a Lie algebroid, $I \\subseteq C^\\infty(X)$ a smooth divisor ideal and $\\mathcal{B} \\to Z_I$ a Lie subalgebroid of $\\mathcal{A}$. Then $\\{[\\mathcal{A}\\:(\\mathcal{B},I)]\\:(\\mathcal{B},I)\\} \\cong \\mathcal{A}$, and similarly when there is a surjective Lie algebroid comorphism from $\\mathcal{B}$ onto $\\mathcal{A}$ we have $[\\{\\mathcal{A}\\:(\\mathcal{B},I)\\}\\:(\\mathcal{B},I)] \\cong \\mathcal{A}$.\n\\end{prop}\n\\subsubsection{Examples}\n\\label{sec:examplesmodification}\nWe finish the discussion of modifications by considering some examples. See also \\cite{Melrose95} for further examples when $I = I_Z$, which are of interest in geometric analysis.\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = TX$ and $\\mathcal{B} = 0_Z$. Then $[\\mathcal{A}\\:\\mathcal{B}] = \\mathcal{B}_Z$, the zero tangent bundle associated to $Z$ (see \\autoref{exa:zvx}).\n\\end{exa}\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = TX$ and $\\mathcal{B} = TZ$. Then $[\\mathcal{A}\\:\\mathcal{B}] = \\mathcal{A}_Z$, the log-tangent bundle associated to $Z$ (see \\autoref{exa:zvx}).\n\\end{exa}\nThe above example can be extended to cases where $\\mathcal{B} = TF$ for $F$ an involutive distribution on $Z$, for example the kernel of a closed one-form on $Z$. The resulting Lie algebroid $TX(-\\log(Z,F)) := [TX\\:TF]$ occurs in the context of log-Poisson structures (c.f.\\ \\cite{KlaasseLanius18,GualtieriLiPelayoRatiu17}).\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = \\mathcal{A}_Z$ and $\\mathcal{B} = TZ$ with surjective bundle morphism $\\rho_\\mathcal{A}|_Z\\colon \\mathcal{A}|_Z \\to \\mathcal{B}$. Then $\\rho_\\mathcal{A}$ is a Lie algebroid comorphism and $\\{\\mathcal{A}_Z\\:\\mathcal{B}\\} = TX$.\n\\end{exa}\n\\begin{exa} Let $(X,Z)$ be a log pair and consider $\\mathcal{A} = \\mathcal{A}_Z$ and $\\mathcal{B} = 0_Z$. Then $[\\mathcal{A}\\:\\mathcal{B}] = \\mathcal{C}_Z$, the \\emph{scattering tangent bundle} of \\cite{Lanius16,Melrose95}.\n\\end{exa}\nFurther examples obtained using elliptic divisor ideals can be found in \\cite{KlaasseLanius18}.\n\n\n\\section{Poisson structures of divisor-type}\n\\label{sec:poissondivtype}\nIn this section we discuss the notion of an $\\mathcal{A}$-Poisson structure (c.f.\\ \\cite{CannasDaSilvaWeinstein99}), their interaction with divisors, and the process of lifting using base-preserving Lie algebroid morphisms. Most $\\mathcal{A}$-Poisson structures we will encounter will be generically nondegenerate, or generically regular in a precise sense. At first reading, the reader can let $\\mathcal{A} = TX$ in most of what follows.\n\\begin{defn} An \\emph{$\\mathcal{A}$-Poisson structure} is an $\\mathcal{A}$-bivector $\\pi_\\mathcal{A}\\in \\Gamma(\\wedge^2\\mathcal{A})$ with $[\\pi_\\mathcal{A},\\pi_\\mathcal{A}]_{\\mathcal{A}} = 0$.\n\\end{defn}\nWe denote the space of such $\\mathcal{A}$-bivectors by ${\\rm Poiss}(\\mathcal{A})$, and set ${\\rm Poiss}(X) := {\\rm Poiss}(TX)$. Of particular interest to us are those Poisson bivectors that are \\emph{generically nondegenerate}, i.e.\\ for which $\\pi^\\sharp_{\\mathcal{A}}\\colon \\mathcal{A}^* \\to \\mathcal{A}$ is nondegenerate on an open dense subset of $X$. As for Poisson structures, an $\\mathcal{A}$-Poisson structure induces an $\\mathcal{A}$-Lie algebroid structure on the dual bundle $\\mathcal{A}^* \\to X$.\n\\begin{defn}\\label{defn:apoissonalgebroid} Let $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$ be given. The \\emph{$\\mathcal{A}$-Poisson algebroid} $\\mathcal{A}^*_{\\pi_\\mathcal{A}}$ consists of the bundle $\\mathcal{A}^*$ with $\\mathcal{A}$-anchor map $\\pi_\\mathcal{A}^\\sharp\\colon \\mathcal{A}^* \\to \\mathcal{A}$, whose Lie bracket on $\\Gamma(\\mathcal{A}^*_{\\pi_{\\mathcal{A}}})$ is given by\n\t%\n\t\\begin{equation*}\n\t[v,w]_{\\mathcal{A}^*_{\\pi_\\mathcal{A}}} = \\mathcal{L}_{\\pi_\\mathcal{A}^\\sharp v} w - \\mathcal{L}_{\\pi_\\mathcal{A}^\\sharp w} v - d_{\\mathcal{A}} \\pi_{\\mathcal{A}}(v,w), \\qquad v,w \\in \\Gamma(\\mathcal{A}^*_{\\pi_{\\mathcal{A}}}).\n\t\\end{equation*}\n\\end{defn}\nPoisson structures can be related to the divisors of \\autoref{sec:divonmanifolds} as follows. Denote the space of $\\mathcal{A}$-bivectors by $\\mathfrak{X}^2(\\mathcal{A}) = \\Gamma(\\wedge^2 \\mathcal{A})$ (\\emph{not} $\\Gamma(\\wedge^2 T\\mathcal{A})$), so that we have $\\mathfrak{X}^2(X) = \\mathfrak{X}^2(TX)$, although no confusion should arise. The \\emph{Pfaffian} of $\\pi_\\mathcal{A}\\in \\mathfrak{X}^2(\\mathcal{A}^{2n})$ is the section $\\wedge^n \\pi_\\mathcal{A} \\in \\Gamma(\\det(\\mathcal{A}))$. An $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}}$ is called \\emph{nondegenerate} if $\\pi_{\\mathcal{A}}^\\sharp\\colon \\mathcal{A}^* \\to \\mathcal{A}$ is an isomorphism, whose existence forces the rank of $\\mathcal{A}$ to be even. Alternatively, one can demand the Pfaffian $\\wedge^n \\pi_\\mathcal{A}$ to be nowhere vanishing, and similarly for $\\wedge^m \\pi_{\\mathcal{A}}$ regarding $2m$-regularity.\n\\begin{defn}\\label{defn:adivisortype} An $\\mathcal{A}$-Poisson structure $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$ is of \\emph{$m$-divisor-type} for some $m \\geq 0$ if $\\wedge^{m+1} \\pi_{\\mathcal{A}} \\equiv 0$ and there exists a line subbundle $K \\subseteq \\wedge^{2m} \\mathcal{A}$ such that $(K, \\wedge^{m} \\pi_\\mathcal{A})$ is a divisor. Denote the divisor of $\\pi_\\mathcal{A}$ by ${\\rm div}(\\pi_\\mathcal{A})$ and the associated divisor ideal by $I_{\\pi_\\mathcal{A}} \\subseteq C^\\infty(X)$.\n\\end{defn}\nNote that if $\\pi_\\mathcal{A}$ is of $m$-divisor-type, it generically has rank $2m$, and if $\\pi_{\\mathcal{A}}$ specifies the trivial divisor, then $\\pi_{\\mathcal{A}}$ is $2m$-regular. In the above definition, when $2m = {\\rm rank}(\\mathcal{A})$, we will simply say that $\\pi_\\mathcal{A}$ is of \\emph{divisor-type}, and then the line bundle $K = \\det(\\mathcal{A})$ can be used.\n\\begin{rem}\\label{rem:apoissontriangbialgebroid} An $\\mathcal{A}$-Poisson structure defines a triangular Lie bialgebroid $(\\mathcal{A}, \\mathcal{A}^*_{\\pi_\\mathcal{A}})$ \\cite{KosmannSchwarzbach95,KosmannSchwarzbachLaurentGengoux05,LiuWeinsteinXu97,MackenzieXu94}: in general, a pair of Lie algebroids $(\\mathcal{A},\\mathcal{A}^*)$ is a Lie bialgebroid if for all $v, w \\in \\Gamma(\\mathcal{A})$\n\t%\n\t\\begin{equation*}\n\td_{\\mathcal{A}^*} [v,w]_\\mathcal{A} = \\mathcal{L}_{v} d_{\\mathcal{A}^*} w - \\mathcal{L}_w d_{\\mathcal{A}^*} v.\n\t\\end{equation*}\n\t%\n\tAny Lie bialgebroid $(\\mathcal{A},\\mathcal{A}^*)$ induces a Poisson structure $\\pi \\in {\\rm Poiss}(X)$ with $\\pi^\\sharp = \\rho_{\\mathcal{A}} \\circ \\rho_{\\mathcal{A}^*}^*$. Not every Lie bialgebroid is triangular (yet $\\pi = \\rho_{\\mathcal{A}}(\\pi_{\\mathcal{A}})$ if it is), and the maps $\\pi_{\\mathcal{A}}^\\sharp \\colon \\mathcal{A}^* \\to \\mathcal{A}$ and $\\rho_{\\mathcal{A}^*}^* \\colon T^*X \\to \\mathcal{A}^*$ are Lie algebroid morphisms. These types of structures are also considered in \\cite{Lanius16,Lanius18}, for Lie algebroids whose isomorphism locus is the complement of a hypersurface.\n\\end{rem}\n\\begin{rem} Given $\\pi_{\\mathcal{A}} \\in {\\rm Poiss}(\\mathcal{A})$ we obtain a (singular) $\\mathcal{A}$-distribution $D_{\\pi_\\mathcal{A}} = \\pi_\\mathcal{A}^\\sharp(\\mathcal{A}^*)$. Regularity of $\\pi_\\mathcal{A}$ is the same as regularity of $D_{\\pi_\\mathcal{A}}$, and then $\\det(D_{\\pi_\\mathcal{A}}) = K$, with $K \\subseteq \\wedge^{2m} \\mathcal{A}$.\n\\end{rem}\n\\begin{rem} If $\\pi_\\mathcal{A}$ is of $m$-divisor-type, it is generically regular of rank $2m$. This is not an if and only if unless $2m = \\dim(X)$: being of $m$-divisor-type for $2m < \\dim(X)$ further demands that the way in which $\\pi_{\\mathcal{A}}$ degenerates is controlled in a precise sense. See also \\autoref{sec:almostregularpoisson}.\n\\end{rem}\nGiven a divisor ideal $I \\subseteq C^\\infty(X)$, we say $\\pi_\\mathcal{A}$ is of \\emph{($m$-)$I$-divisor-type} if $I_{\\pi_\\mathcal{A}} = I$, noting that we must specify both the divisor ideal $I$, and the generic rank $2m$. Given a divisor ideal $I$, we denote by $\\mathfrak{X}^2_I(\\mathcal{A}) \\subseteq \\mathfrak{X}^2(\\mathcal{A})$ the space of divisor $\\mathcal{A}$-bivectors $\\pi_\\mathcal{A}$ such that $I_{\\pi_\\mathcal{A}} = I$. We denote by ${\\rm Poiss}_I(\\mathcal{A})$ the space of $\\mathcal{A}$-Poisson structures of $I$-divisor-type, and by ${\\rm Poiss}_{I,m}(\\mathcal{A})$ the space of $\\mathcal{A}$-Poisson structures of $m$-$I$-divisor-type.\n\\begin{rem} One of our main interests lies in nondegenerate $\\mathcal{A}$-Poisson structures, as we then have access to symplectic techniques (see \\autoref{sec:symplecticliealgebroids}). Indeed, the goal of the lifting process described in Section \\ref{sec:liftingpoisson} is to find a Lie algebroid $\\mathcal{A} \\to X$ for which a given $\\pi \\in {\\rm Poiss}(X)$ can be viewed as a nondegenerate $\\mathcal{A}$-Poisson structure.\n\\end{rem}\nA function $f \\in C^\\infty(X)$ is said to be \\emph{$\\pi_{\\mathcal{A}}$-Casimir} if $\\pi_\\mathcal{A}^\\sharp(d_\\mathcal{A} f) = 0$ (alternatively, $[f,\\pi_{\\mathcal{A}}]_\\mathcal{A} = 0$ using the $\\mathcal{A}$-Schouten bracket). Given an $\\mathcal{A}$-Poisson structure $\\pi_\\mathcal{A}$ of $m$-divisor-type and a $\\pi_\\mathcal{A}$-Casimir function $f$, we can form a new $\\mathcal{A}$-Poisson structure $\\pi'_\\mathcal{A} = f \\cdot \\pi_\\mathcal{A}$. If the zero set of $f$ is nowhere dense, it specifies a divisor $(\\underline{\\mathbb{R}},f)$, so as $\\wedge^{m} \\pi'_\\mathcal{A} = f^m \\cdot \\wedge^m \\pi_\\mathcal{A}$ we conclude that $\\pi'_\\mathcal{A}$ is again of $m$-divisor-type. Moreover, $I_{\\pi'_\\mathcal{A}} =  \\langle f^m \\rangle \\cdot I_{\\pi_\\mathcal{A}}$. See also \\cite[Lemma 3.3]{AndroulidakisZambon17} and \\autoref{sec:almostregularpoisson}.\n\\begin{rem} When $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$ has maximal rank equal to $2$ it follows that $f \\cdot \\pi_\\mathcal{A}$ is $\\mathcal{A}$-Poisson for \\emph{any} function $f \\in C^\\infty(X)$ (as is known for $\\mathcal{A} = TX$): we have $[f\\cdot \\pi_\\mathcal{A}, f \\cdot \\pi_\\mathcal{A}]_\\mathcal{A} = f^2 [\\pi_\\mathcal{A}, \\pi_\\mathcal{A}]_\\mathcal{A} + 2 f \\pi_\\mathcal{A} \\wedge \\pi_\\mathcal{A}^\\sharp(d_\\mathcal{A} f)$, where the latter term vanishes as $\\pi_\\mathcal{A} \\wedge v = 0$ for all $v \\in {\\rm im}(\\pi^\\sharp_\\mathcal{A})$. This can be used to construct examples of $\\mathcal{A}$-Poisson structures of higher divisor-type.\n\\end{rem}\nGiven an $\\mathcal{A}$-Poisson structure, we obtain its \\emph{$\\mathcal{A}$-Hamiltonian vector fields} as the image of the map $f \\mapsto \\pi_\\mathcal{A}^\\sharp(d_\\mathcal{A} f) =: v_{\\mathcal{A}, f} \\in \\mathfrak{X}(\\mathcal{A})$. Similarly we have the \\emph{$\\mathcal{A}$-Poisson vector fields}, i.e.\\ those $v_\\mathcal{A} \\in \\mathfrak{X}(\\mathcal{A})$ such that $\\mathcal{L}_{v_\\mathcal{A}} \\pi_{\\mathcal{A}} = 0$, using the $\\mathcal{A}$-Lie derivative. Define the spaces ${\\rm Ham}(\\pi_\\mathcal{A}) \\subseteq {\\rm Poiss}(\\pi_\\mathcal{A}) \\subseteq \\mathfrak{X}(\\mathcal{A})$ as for Poisson structures. Given $f \\in C^\\infty(X)$ we have $\\pi_{\\mathcal{A}}^\\sharp(d_{\\mathcal{A}} f) = \\pi_{\\mathcal{A}}^\\sharp(\\rho_{\\mathcal{A}}^*(df))$, so that by setting $\\pi := \\rho_\\mathcal{A}(\\pi_\\mathcal{A})$, we see from the relation $\\pi^\\sharp = \\rho_{\\mathcal{A}} \\circ \\pi_{\\mathcal{A}}^\\sharp \\circ \\rho_{\\mathcal{A}}^*$ that $\\rho_\\mathcal{A}(v_{\\mathcal{A},f}) = V_f$. Thus the $\\mathcal{A}$-Hamiltonian vector fields for $\\pi_\\mathcal{A}$ surject onto the Hamiltonian vector fields for $\\pi$.\n\\begin{rem}\\label{rem:productapoisson} Let $(X,\\mathcal{A},\\pi_\\mathcal{A})$ and $(X',\\mathcal{A}',\\pi_{\\mathcal{A}'})$ be two $\\mathcal{A}$-Poisson manifolds. Then their product $(X \\times X', \\mathcal{A} \\oplus \\mathcal{A}', \\pi_{\\mathcal{A}} + \\pi_{\\mathcal{A}'})$ is also $\\mathcal{A}$-Poisson, where $\\mathcal{A} \\oplus \\mathcal{A}' \\to X \\times X'$ is the direct product of Lie algebroids (see \\cite{HigginsMackenzie90}), and both projections are $\\mathcal{A}$-Poisson maps. Thus the class of $\\mathcal{A}$-Poisson structures of divisor-type is closed under products, using the external tensor product of divisors (see \\autoref{defn:divdirectstum}). This is noted in \\cite{AndroulidakisZambon17} for $\\mathcal{A} = TX$ using \\autoref{cor:divtypealmostreg}.\n\\end{rem}\n\\subsection{Degeneraci loci}\nWe discuss degeneraci loci of $\\mathcal{A}$-Poisson structures $\\pi_\\mathcal{A}$, as in \\cite{Pym13}. Given $\\pi_{\\mathcal{A}} \\in \\mathfrak{X}^2(\\mathcal{A}^n)$ with bundle map $\\pi_\\mathcal{A}^\\sharp\\colon \\mathcal{A}^* \\to \\mathcal{A}$, we have both $\\det(\\pi_\\mathcal{A}^\\sharp) \\in \\Gamma(\\det(\\mathcal{A})^2)$ and $\\wedge^n \\pi_\\mathcal{A} \\in \\Gamma(\\det(\\mathcal{A}))$, which are related by $\\det(\\pi_\\mathcal{A}^\\sharp) = \\wedge^n \\pi_\\mathcal{A} \\otimes \\wedge^n \\pi_\\mathcal{A}$. Considering the definition for Lie algebroids, we could define the degeneracy loci of $\\pi_\\mathcal{A}$ using the map $\\pi_\\mathcal{A}^\\sharp$, i.e.\\ as the degeneracy locus of its $\\mathcal{A}$-Poisson algebroid $\\mathcal{A}^*_{\\pi_{\\mathcal{A}}}$. However, $\\pi^\\sharp_\\mathcal{A}$ is skew-symmetric, so its vanishing degree is too high. Instead we will use the Pfaffians $\\wedge^k\\pi_\\mathcal{A} \\in \\Gamma(\\wedge^{2k} \\mathcal{A})$ for $k \\geq 0$. These give rise to maps $\\wedge^k \\pi_\\mathcal{A}\\colon \\Gamma(\\wedge^{2k} \\mathcal{A}^*) \\to C^\\infty(X)$ on sections, and thus define ideals $I_{\\pi_\\mathcal{A},2k}$.\n\\begin{defn} Let $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$. The \\emph{$2k$th degeneracy locus} of $\\pi_\\mathcal{A}$ for $k \\geq 0$ is the subspace $X_{\\pi_\\mathcal{A},2k} \\subseteq X$ of points where ${\\rm rank}(\\pi_{\\mathcal{A}}) \\leq 2k$, defined by the ideal $I_{\\pi_\\mathcal{A},2k+2} \\subseteq C^\\infty(X)$.\n\\end{defn}\nThus the degeneraci loci of $\\mathcal{A}_{\\pi_\\mathcal{A}}^*$ and $\\pi_\\mathcal{A}$ agree as subspaces of $X$, but their ideals do not. Recall that a submanifold $N$ is \\emph{$\\pi$-Poisson} if $\\pi^\\sharp(T_x^* X) \\subseteq T_x N$ for all $x \\in N$. When $N$ is closed this is equivalent to its vanishing ideal $I_N$ being a \\emph{Poisson ideal}, i.e.\\ $\\{I_N,C^\\infty(X)\\}_\\pi \\subseteq I_N$, or to $N$ being $T^*_\\pi X$-invariant. Moreover, note that $I$ being a $\\pi$-Poisson ideal is equivalent to $\\pi^\\sharp(d I) = 0$. When $N$ is $\\pi$-Poisson, it carries a uniquely induced Poisson structure $\\pi_N$ for which the inclusion is a Poisson map. The following is then a consequence of \\autoref{prop:degenlocus}.\n\\begin{prop}\\label{prop:pidegenpipoisson} Each degeneracy locus $X_{\\pi,2k}$ is a $\\pi$-Poisson submanifold if it is smooth.\n\\end{prop}\nFor $\\mathcal{A}$-Poisson structures the notion of a \\emph{$\\pi_{\\mathcal{A}}$-Poisson submanifold} is more involved: it instead is a Lie subalgebroid $(\\varphi,i)\\colon (\\mathcal{B},N) \\hookrightarrow (\\mathcal{A},X)$ with $\\pi_\\mathcal{B} \\in {\\rm Poiss}(\\mathcal{B})$ with $\\varphi_*(\\pi_\\mathcal{B}) = \\pi_\\mathcal{A}$.\nA point $x \\in X$ is $\\pi_{\\mathcal{A}}$-\\emph{regular} if the rank of $\\pi_\\mathcal{A}$ is constant in some open neighbourhood of $x$, and singular otherwise. We denote by $X_{\\pi_\\mathcal{A}, {\\rm reg}} \\subseteq X$ the space of $\\pi_\\mathcal{A}$-regular points, also called the \\emph{regular locus} of $\\pi_\\mathcal{A}$. This is an open dense subspace of $X$, with its complement $X_{\\pi_\\mathcal{A}, {\\rm sing}}$, the \\emph{singular locus} of $\\pi_\\mathcal{A}$, being closed and nowhere dense. Denote by $X_{\\pi_\\mathcal{A}, {\\rm max}} \\subseteq X$ the open subspace where $\\pi_\\mathcal{A}$ has maximal rank. It immediately follows that $X \\backslash X_{\\pi_\\mathcal{A}, {\\rm max}} = X_{\\pi_\\mathcal{A}, 2k-2} = (\\wedge^k \\pi_\\mathcal{A})^{-1}(0)$, where $2k$ is the maximal rank of $\\pi_\\mathcal{A}$. In general we have $X_{\\pi_\\mathcal{A}, {\\rm max}} \\subsetneq X_{\\pi_\\mathcal{A}, {\\rm reg}}$.\n\\subsection{Relation to almost-regularity}\n\\label{sec:almostregularpoisson}\nA Poisson structure $\\pi \\in {\\rm Poiss}(X)$ specifies a singular Lie subalgebroid $\\mathcal{F}_\\pi := \\pi^\\sharp(\\Gamma(T_\\pi^*X))$ of $TX$ consisting of $\\pi$-Hamiltonian vector fields. Following Androulidakis--Zambon \\cite{AndroulidakisZambon17}, we call the Poisson structure $\\pi$ \\emph{almost-regular} if $\\mathcal{F}_\\pi$ is projective. There is a nice criterion for almost-regularity in terms of the $\\pi$-symplectic leaves.\n\\begin{prop}[{\\cite[Theorem 2.8]{AndroulidakisZambon17}}]\\label{prop:almostregulardistr} Let $\\pi \\in {\\rm Poiss}(X)$. Then $\\pi$ is almost-regular if and only if $X_{\\pi, {\\rm max}} \\subseteq X$ is dense and there exists a distribution $D_\\pi \\subseteq TX$ such that $D_{\\pi,x} = T_x L$ for all $x \\in X_{\\pi, {\\rm max}}$, where $L$ is the $\\pi$-symplectic leaf through $x$.\n\\end{prop}\nHence almost-regularity of $\\pi$ depends only on the partition of $X$ into immersed leaves.\n\\begin{rem} As is stressed in \\cite{AndroulidakisZambon17}, in general projectivity of a singular Lie subalgebroid $\\rho_{A}(\\Gamma(\\mathcal{A})) \\subseteq \\Gamma(TX)$ cannot be tested from just the orbits of $\\mathcal{A}$. However, this is possible for those singular Lie subalgebroids arising from Poisson structures due to skew-symmetry of the bivector $\\pi$, which implies that the anchor $\\pi^\\sharp$ of $T^*_\\pi X$ is skew, so that $\\ker(\\pi^\\sharp_x) = {\\rm im}(\\pi^\\sharp_x)^\\circ$.\n\\end{rem}\nBy continuity and density of $X_{\\pi, {\\rm max}}$, the distribution $D_\\pi$ in the above proposition is unique and involutive, and determines a $2k$-dimensional regular foliation by $\\pi$-Poisson submanifolds. Phrased differently, $D_\\pi$ specifies a $2k$-regular Lie algebroid whose orbits are $\\pi$-Poisson submanifolds. Note that $X_{\\pi, {\\rm max}} = X_{\\pi, {\\rm reg}}$ for almost-regular Poisson structures. There is another characterization of almost-regularity, phrased in terms of $\\pi$ itself. Let ${\\rm Poiss}_{2k}(X) \\subseteq {\\rm Poiss}(X)$ be the space of Poisson structures of maximal rank $2k$, i.e.\\ for which $X_{\\pi,{\\rm max}} = X \\backslash X_{2k-2}$.\n\\begin{prop}[{\\cite[Proposition 2.11]{AndroulidakisZambon17}}]\\label{prop:almostregularline} Let $\\pi \\in {\\rm Poiss}_{2k}(X)$. Then $\\pi$ is almost-regular if and only if $X_{\\pi, {\\rm max}}$ is dense and there exists a line bundle $K \\subseteq \\wedge^{2k} TX$ such that $\\wedge^k \\pi \\in \\Gamma(K)$.\n\\end{prop}\nThe above distribution $D_\\pi$ and line bundle $K$ are related via $\\det(D_\\pi) = \\wedge^k D_\\pi = K$. Note that because for $\\pi$ as in \\autoref{prop:almostregularline} we have $X_{\\pi,{\\rm max}} = X_{\\pi,{\\rm reg}} = X \\backslash X_{2k-2} = (\\wedge^{k} \\pi)^{-1}(0)$, so that density of this subspace amounts to saying that the pair $(K,\\wedge^k \\pi)$ is a divisor on $X$.\n\\begin{rem} In work of Lanius \\cite{Lanius16two,Lanius16,Lanius17}, specifically for computing Poisson cohomology, the \\emph{rigged algebroid} $\\mathcal{R}_\\pi$ of a Poisson structure $\\pi$ plays a large role. It is defined as the almost-injective Lie algebroid inducing $\\mathcal{F}_\\pi = \\pi^\\sharp(\\Gamma(T^*X))$. In the joint paper \\cite{KlaasseLanius18} we systematically develop its use in computing the Poisson cohomology of almost-regular Poisson structures.\n\\end{rem}\nTogether with \\autoref{defn:adivisortype} we obtain the following dictionary between concepts.\n\\begin{cor}\\label{cor:divtypealmostreg} Let $\\pi \\in {\\rm Poiss}(X)$ be given. Then the following are equivalent:\n\t%\n\t\\begin{itemize}\n\t\\item $\\pi$ is of $m$-divisor-type for some $m \\geq 0$;\n\t\\item $\\pi$ is almost-regular;\n\t\\item The rigged algebroid $\\mathcal{R}_\\pi$ exists.\n\t\\end{itemize}\n\\end{cor}\nBy defining almost-regularity for $\\mathcal{A}$-Poisson structures as projectivity of $\\pi^\\sharp_{\\mathcal{A}}(\\Gamma(\\mathcal{A}^*))$, and its \\emph{$\\mathcal{A}$-rigged algebroid} $\\mathcal{R}_{\\pi_{\\mathcal{A}}}$ via $\\Gamma(\\mathcal{R}_{\\pi_{\\mathcal{A}}}) \\cong \\pi^\\sharp_{\\mathcal{A}}(\\Gamma(\\mathcal{A}^*))$ \\cite{KlaasseLanius18}, \\autoref{cor:divtypealmostreg} is seen to hold for $\\mathcal{A}$-Poisson structures for any Lie algebroid $\\mathcal{A}$, after adapting the proof of \\cite[Proposition 2.11]{AndroulidakisZambon17}.\n\\subsection{Lifting Poisson structures}\n\\label{sec:liftingpoisson}\nWe next introduce the process of lifting $\\mathcal{A}$-Poisson structures through base-preserving Lie algebroid morphisms (see also the discussion in \\cite{CavalcantiKlaasse18}).\n\\begin{defn}\\label{defn:poissonalift} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ be a Lie algebroid morphism. An $\\mathcal{A}'$-bivector $\\pi_{\\mathcal{A}'} \\in \\Gamma(\\wedge^2 \\mathcal{A}')$ is \\emph{$\\mathcal{A}$-liftable} if there exists an $\\mathcal{A}$-bivector $\\pi_{\\mathcal{A}} \\in \\Gamma(\\wedge^2 \\mathcal{A})$ such that $\\varphi(\\pi_{\\mathcal{A}}) = \\pi_{\\mathcal{A}'}$.\n\\end{defn}\nIn the above situation we again call $\\pi_{\\mathcal{A}}$ the \\emph{$\\mathcal{A}$-lift} of $\\pi_{\\mathcal{A}'}$, and say that $\\pi_{\\mathcal{A}'}$ is of \\emph{$\\mathcal{A}$-type}. We will denote the space of $\\mathcal{A}$-liftable bivectors by $\\mathfrak{X}_\\mathcal{A}^2(X) := \\rho_\\mathcal{A}(\\mathfrak{X}^2(\\mathcal{A})) \\subset \\mathfrak{X}^2(X)$, and the space of $\\mathcal{A}$-liftable Poisson structures by ${\\rm Poiss}_{\\mathcal{A}}(X) \\subseteq {\\rm Poiss}(X)$.\n\\begin{rem} In \\cite[Definition 2.16]{Lanius16} this notion is considered using slightly different terminology: relating it to ours, there $\\pi \\in {\\rm Poiss}(X)$ is called $\\mathcal{A}$-Poisson if it is of $\\mathcal{A}$-type. It seems preferable to describe Poisson structures by their divisor (c.f.\\ \\autoref{defn:adivisortype}) instead of to which Lie algebroid they can be (nondegenerately) lifted, as the latter need not be unique.\n\\end{rem}\nThe $\\mathcal{A}$-lift of a $\\mathcal{A}'$-Poisson structure of $\\mathcal{A}$-type need not necessarily be $\\mathcal{A}$-Poisson (consider $\\mathcal{A}$ with trivial anchor and $\\pi \\equiv 0$). However, this is true when $\\varphi$ is of divisor-type, as will be the case for us (moreover, the $\\mathcal{A}$-lift will then be unique). More generally this holds for almost-injective $\\varphi$. We have $\\pi_{\\mathcal{A}'}^\\sharp = \\varphi \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\varphi^*$ as maps, summarized in the following diagram.\n\\begin{center}\n\t\\begin{tikzpicture}\n\t%\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t\\mathcal{A}^* & \\mathcal{A} \\\\ \\mathcal{A}'{}^* & \\mathcal{A}' \\\\};\n\t\\path[-stealth]\n\t(m-1-1) edge node [above] {$\\pi_\\mathcal{A}^\\sharp$} (m-1-2)\n\t(m-2-1) edge node [left] {$\\varphi^*$} (m-1-1)\n\t(m-2-1) edge node [above] {$\\pi_{\\mathcal{A}'}^\\sharp$} (m-2-2)\n\t(m-1-2) edge node [right] {$\\varphi$} (m-2-2);\n\t\\end{tikzpicture}\n\\end{center}\n\\begin{defn}\\label{defn:apoissonmap} An \\emph{$\\mathcal{A}$-Poisson map} $(\\varphi,f)\\colon (\\mathcal{A},X,\\pi_\\mathcal{A}) \\to (\\mathcal{B},X',\\pi_\\mathcal{B})$ is a Lie algebroid morphism between Lie algebroids with $\\mathcal{A}$-Poisson structures such that $\\varphi(\\pi_\\mathcal{A}) = \\pi_\\mathcal{B}$.\n\\end{defn}\nAs for Poisson maps, it is not true that any $\\mathcal{A}$-Poisson structure can be pushed forward to an $\\mathcal{A}'$-Poisson structure along a given Lie algebroid morphism $(\\varphi,f)\\colon \\mathcal{A} \\to \\mathcal{A}'$. However, this is always possible along base-preserving morphisms. In particular, given an $\\mathcal{A}$-Poisson structure, the bivector $\\pi := \\rho_{A}(\\pi_{\\mathcal{A}})$ is always Poisson. The lifting condition of an $\\mathcal{A}'$-Poisson structure to an $\\mathcal{A}$-Poisson structure is exactly that $(\\varphi,{\\rm id}_X)\\colon (\\mathcal{A},\\pi_\\mathcal{A}) \\to (\\mathcal{A}',\\pi_{\\mathcal{A}'})$ is an $\\mathcal{A}$-Poisson map. If $\\pi_{\\mathcal{A}'}$ has a nondegenerate $\\mathcal{A}$-Poisson lift $\\pi_\\mathcal{A}$, we say $\\pi_{\\mathcal{A}'}$ is of \\emph{nondegenerate $\\mathcal{A}$-type}, and similarly for other types (e.g.\\ for regular ones). We often omit writing the Lie algebroid morphism $\\varphi$ (as typically it is induced by an inclusion of modules), and always use the anchor map $\\rho_{\\mathcal{A}}$ when considering $\\mathcal{A}$-lifts of bivectors on $TX$. When $\\pi_{\\mathcal{A}}$ nondegenerately lifts $\\pi_{\\mathcal{A}'}$, we will call its inverse $\\omega_{\\mathcal{A}} := \\pi_\\mathcal{A}^{-1} \\in \\Omega^2(\\mathcal{A})$ the \\emph{dual $\\mathcal{A}$-form} to $\\pi_{\\mathcal{A}'}$. This is an $\\mathcal{A}$-symplectic structure in the sense of \\autoref{sec:symplecticliealgebroids}, making $\\mathcal{A}$ into a symplectic Lie algebroid \\cite{NestTsygan01}.\n\nThus the process of lifting constitutes the existence of a base-preserving $\\mathcal{A}$-Poisson map. In terms of $\\mathcal{A}$-Poisson algebroids, we obtain the following analogue of the well-known fact that a Poisson map induces a Lie algebroid comorphism between the respective Poisson algebroids.\n\\begin{prop}\\label{prop:apoissoncomorphism} Let $(\\varphi,f)\\colon (\\mathcal{A},X,\\pi_\\mathcal{A}) \\to (\\mathcal{B},X',\\pi_\\mathcal{B})$ be an $\\mathcal{A}$-Poisson map. Then there is an induced $\\mathcal{A}$-Lie algebroid comorphism $(\\varphi;\\varphi,f)\\colon (\\mathcal{A}^*_{\\pi_\\mathcal{A}},\\mathcal{A},X) \\dashedrightarrow (\\mathcal{B}^*_{\\pi_\\mathcal{B}},\\mathcal{B},X')$.\n\\end{prop}\nThe lifting process interacts with Poisson structures of divisor-type as follows.\n\\begin{prop}\\label{prop:apoissondivtypelifts} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ be a Lie algebroid morphism of divisor-type, and let $\\pi_{\\mathcal{A}'}$ be an $\\mathcal{A}'$-Poisson structure of $m$-divisor-type for some $m \\geq 0$ that is also of $\\mathcal{A}$-type. Then its unique $\\mathcal{A}$-lift $\\pi_\\mathcal{A}$ is of $m$-divisor-type, and ${\\rm div}(\\pi_{\\mathcal{A}'}) = {\\rm div}(\\pi_\\mathcal{A}) \\cdot {\\rm div}(\\varphi)$ so that $I_{\\pi_{\\mathcal{A}'}} = I_{\\pi_{\\mathcal{A}}} \\cdot I_\\varphi$.\n\\end{prop}\n\\bp Let $\\pi_\\mathcal{A}$ be an $\\mathcal{A}$-lift of $\\pi_{\\mathcal{A}'}$, so that $(\\wedge^2 \\varphi)(\\pi_\\mathcal{A}) = \\pi_{\\mathcal{A}'}$. We have ${\\rm div}(\\pi_{\\mathcal{A}'}) = (K, \\wedge^m \\pi_{\\mathcal{A}'})$ for some line bundle $K \\subseteq \\wedge^{2m} \\mathcal{A}'$. As $\\wedge^{2m} \\varphi\\colon \\wedge^{2m} \\mathcal{A} \\to \\wedge^{2m} \\mathcal{A}'$, we find a line bundle $K_\\mathcal{A} \\subseteq \\wedge^{2m} \\mathcal{A}$ for which $\\wedge^{2m}\\varphi\\colon K_\\mathcal{A} \\to K$ and $\\wedge^m \\pi_\\mathcal{A} \\in \\Gamma(K_\\mathcal{A})$. If $\\pi_\\mathcal{A}$ vanishes then so must $\\pi_{\\mathcal{A}'}$, so that $(K_\\mathcal{A}, \\wedge^m \\pi_\\mathcal{A})$ is a divisor, ${\\rm div}(\\pi_{\\mathcal{A}})$, as a subset of a nowhere dense set is nowhere dense. The statement that ${\\rm div}(\\pi_{\\mathcal{A}'}) = {\\rm div}(\\pi_\\mathcal{A}) \\cdot {\\rm div}(\\varphi)$ then follows from the relation $\\pi_{\\mathcal{A}'}^\\sharp = \\varphi \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\varphi^*$.\n\\end{proof}\nThis proposition makes precise that the process of lifting makes $\\mathcal{A}$-Poisson structures less degenerate, as their degeneracies are absorbed in that of the Lie algebroid morphism (hence, in the Lie algebroid it is lifted to). Moreover, it says that $\\pi_{\\mathcal{A}'}$ is of $m$-regular $\\mathcal{A}$-type if and only if ${\\rm div}(\\pi_{\\mathcal{A}'}) = {\\rm div}(\\varphi)$, once $\\pi_{\\mathcal{A}'}$ is of $\\mathcal{A}$-type (this includes being of nondegenerate $\\mathcal{A}$-type. There is a similar statement for not necessarily base-preserving $\\mathcal{A}$-Poisson maps.\n\\begin{lem} Let $(\\varphi,f)\\colon (\\mathcal{A}^{2n},X,\\pi_{\\mathcal{A}}) \\to (\\mathcal{B}^{2n},X',\\pi_{\\mathcal{B}})$ be an $\\mathcal{A}$-Poisson map between Lie algebroids with $\\mathcal{A}$-Poisson structures of divisor-type. Then $I_{\\pi_{\\mathcal{A}}}$ divides $f^*I_{\\pi_{\\mathcal{B}}}$.\n\\end{lem}\n\\bp As ${\\rm rank}(\\mathcal{A}) = {\\rm rank}(\\mathcal{B})$, the fact that $\\varphi(\\pi) = \\sigma$ implies that $\\det(\\varphi)(\\pi_{\\mathcal{A}}^n) = \\pi_{\\mathcal{B}}^n$. Take local volume forms $\\mu_\\mathcal{A}, \\mu_\\mathcal{B}$ for $\\mathcal{A}$ and $\\mathcal{B}$. For certain functions $g \\in C^\\infty(X)$, $h \\in C^\\infty(X')$:\n\\begin{equation*}\n\t\\langle\\pi_\\mathcal{A}^n,\\mu_\\mathcal{A}\\rangle = g \\text{, so } I_{\\pi_{\\mathcal{A}}} = \\langle g \\rangle, \\qquad \\text{and} \\qquad \\langle\\pi_{\\mathcal{B}}^n,\\mu_\\mathcal{B}\\rangle = h \\text{, so } I_{\\pi_{\\mathcal{B}}} = \\langle h \\rangle.\n\\end{equation*}\nNote that $\\varphi^* \\mu_\\mathcal{B} = w \\mu_\\mathcal{A}$ for some nonnegative function $w \\in C^\\infty(X)$. Thus $I_{\\pi_{\\mathcal{A}}} \\subseteq f^*I_{\\pi_{\\mathcal{B}}}$, as\n\\begin{equation*}\nf^*(h) = f^*\\langle\\pi_{\\mathcal{B}},\\mu_\\mathcal{B}\\rangle = f^*\\langle\\det(\\varphi)(\\pi_{\\mathcal{A}}^n),\\mu_\\mathcal{B}\\rangle = \\langle\\pi_\\mathcal{A}^n, \\varphi^*\\mu_\\mathcal{B}\\rangle = \\langle\\pi_\\mathcal{A}^n, w \\mu_\\mathcal{A}\\rangle = w \\langle \\pi_\\mathcal{A}^n, \\mu_\\mathcal{A}\\rangle = w g.\\qedhere\n\\end{equation*}\n\\end{proof}\nThe above lemma is of particular interest in the standard case when $\\mathcal{A} = TX$ and $\\mathcal{B} = TX'$.\nFor clarity, in the specific case that $\\mathcal{A}' = TX$, \\autoref{prop:apoissondivtypelifts} says that a Poisson structure $\\pi \\in {\\rm Poiss}(X)$ of divisor-type $I$ can only admit nondegenerate $\\mathcal{A}$-lifts to Lie algebroids satisfying $I_\\mathcal{A} = I$.\nWe can iterate the lifting procedure as follows, whose proof is immediate.\n\\begin{prop}\\label{prop:liftiterate} Let $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ and $(\\varphi',{\\rm id}_X)\\colon \\mathcal{A}' \\to \\mathcal{A}''$ be Lie algebroid morphisms, and $\\pi_{\\mathcal{A}''} \\in \\rm{Poiss}(\\mathcal{A}'')$. Then if $\\pi_{\\mathcal{A}''}$ is of $\\mathcal{A}$-type, it is of $\\mathcal{A}'$-type. If $\\pi_\\mathcal{A}$ is an $\\mathcal{A}$-Poisson lift of $\\pi_{\\mathcal{A}''}$, then $\\pi_{\\mathcal{A}'} := \\varphi(\\pi_\\mathcal{A})$ is an $\\mathcal{A}'$-Poisson lift of $\\pi_{\\mathcal{A}''}$ which itself is of $\\mathcal{A}$-type.\n\\end{prop}\nConsequently, an $\\mathcal{A}'$-Poisson structure $\\pi_{\\mathcal{A}'}$ being of $\\mathcal{A}$-type is the same thing as its underlying Poisson structure $\\pi := \\rho_{\\mathcal{A}'}(\\pi_{\\mathcal{A}'})$ being of $\\mathcal{A}$-type. Let $\\pi_{\\mathcal{A}'}$ be a nondegenerate $\\mathcal{A}'$-Poisson structure. Then any $\\mathcal{A}$-Poisson map $(\\varphi,f)\\colon (\\mathcal{A},\\pi_\\mathcal{A}) \\to (\\mathcal{A}',\\pi_{\\mathcal{A}'})$ has to be a Lie algebroid submersion, i.e.\\ the morphism $\\varphi$ must be fiberwise surjective. This implies the following.\n\\begin{prop}\\label{prop:liftingnondegiso} Let $\\pi \\in {\\rm Poiss}(X)$ be of nondegenerate $\\mathcal{A}' $-type and let  $(\\varphi,{\\rm id}_X)\\colon \\mathcal{A} \\to \\mathcal{A}'$ be a morphism between Lie algebroids of the same rank. Then the following are equivalent:\n\t%\n\t\\begin{itemize}\n\t\\item $\\varphi$ is an isomorphism;\n\t\\item $\\pi$ is of $\\mathcal{A}$-type;\n\t\\item $\\pi$ is of nondegenerate $\\mathcal{A}$-type.\n\t\\end{itemize}\n\\end{prop}\nAs a consequence, once one has lifted $\\pi$ to being nondegenerate, one cannot (meaningfully) lift further. This makes sense, as the lifting process is meant to desingularize the Poisson structure, and nondegenerate ones are maximally nonsingular. This is consistent with \\autoref{prop:apoissondivtypelifts}, as isomorphisms specify trivial divisors. Moreover, we see that $\\pi_{\\mathcal{A}'}$ is of nondegenerate $\\mathcal{A}$-type if and only if the underlying Poisson structure $\\pi = \\rho_{\\mathcal{A}'}(\\pi_{\\mathcal{A}'})$ is.\n\\begin{rem}\\label{rem:multiplelifts} A Poisson structure can lift nondegenerately to multiple non-isomorphic Lie algebroids of (necessarily) the same divisor-type. The orbits of these Lie algebroids will specify the same partition of $X$ into leaves. For some consequences of this phenomenon, see \\cite{KlaasseLanius18}.\n\\end{rem}\n\\begin{exa}\\label{exa:mregularapoisson} Let $\\pi_\\mathcal{A}$ be an $m$-regular $\\mathcal{A}$-Poisson structure. Then its image $D_{\\pi_\\mathcal{A}} = \\pi_\\mathcal{A}^\\sharp(\\mathcal{A}^*)$ is a regular $\\mathcal{A}$-distribution. By \\autoref{exa:adistribution}, $D_{\\pi_{\\mathcal{A}}}$ is a Lie subalgebroid of $\\mathcal{A}$ and thus carries an $\\mathcal{A}$-Lie algebroid structure $\\varphi\\colon D_{\\pi_\\mathcal{A}} \\to \\mathcal{A}$. It follows that $\\pi_{\\mathcal{A}}$ is of nondegenerate $D_{\\pi_\\mathcal{A}}$-type.\n\\end{exa}\nLet us give another consequence of lifting. Let $\\mathcal{A} \\to X$ be a Lie algebroid and recall that $\\mathcal{F}_\\mathcal{A} = \\rho_{\\mathcal{A}}(\\Gamma(\\mathcal{A}))$ is its induced singular foliation whose leaves are the orbits of $\\mathcal{A}$.\n\\begin{prop}\\label{prop:atypepoissonsubmfd} Let $\\pi \\in {\\rm Poiss}_\\mathcal{A}(X)$. Then the orbits of $\\mathcal{A}$ are $\\pi$-Poisson submanifolds.\n\\end{prop}\n\\bp The lifting condition $\\rho_{\\mathcal{A}}(\\pi_{\\mathcal{A}}) = \\pi$ equivalently reads $\\pi^\\sharp = \\rho_{\\mathcal{A}} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{\\mathcal{A}}^*$. As a submanifold $N$ is $\\pi$-Poisson if $\\pi^\\sharp(T^*_x X) \\subseteq T_x N$ for all $x \\in N$, we note that this condition pointwise becomes $\\rho_{\\mathcal{A},x}(\\mathcal{B}_x) \\subseteq T_x N$ for the subspace $\\mathcal{B}_x := \\pi_{\\mathcal{A},x}^\\sharp(\\rho_{\\mathcal{A},x}^*(T^*_x X)) \\subseteq \\mathcal{A}_x$. If $N$ is an orbit of $\\mathcal{A}$, then by definition $T_x N = {\\rm im}(\\rho_{\\mathcal{A},x}) \\subseteq T_x X$, so that this condition is certainly satisfied.\n\\end{proof}\nThis gives an intuitive description of the lifting process when $\\mathcal{A}$ is almost-injective. Namely, a choice of lifting algebroid $\\mathcal{A}$ amounts to a ``grouping'' of the $\\pi$-symplectic leaves into orbits of $\\mathcal{A}$. Unless $\\mathcal{A}$ is regular (with injective anchor), there may be multiple non-isomorphic Lie algebroids with the same orbits (cf.\\ \\autoref{rem:multiplelifts}). There is a partial converse to \\autoref{prop:atypepoissonsubmfd} in the regular injective case, in which case $\\mathcal{A}$ is just a regular foliation $T\\mathcal{F} \\subseteq TX$.\n\\begin{prop}\\label{prop:regularpipoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ and $\\mathcal{F}$ be a regular foliation by $\\pi$-Poisson submanifolds, with $D = T\\mathcal{F}$ the associated regular Lie algebroid. Then $\\pi$ is of $D$-type.\n\\end{prop}\n\\bp Let $L$ be a leaf of $\\mathcal{F}$. As it is $\\pi$-Poisson, we have $\\pi_x \\in \\wedge^2 T_x L = \\wedge^2 D_x$ for all $x \\in L$. By regularity there exists $\\pi_D \\in {\\rm Poiss}(D)$ agreeing with $\\pi$ on each leaf, which is a $D$-lift of $\\pi$.\n\\end{proof}\nThis result recovers the fact that a regular Poisson structure is equivalently described as a nondegenerate Poisson bivector for the Lie algebroid of tangencies to its symplectic leaves. The consequences of \\autoref{prop:regularpipoissonlift} for almost-regular Poisson structures (i.e., those of $m$-divisor-type), in light of \\autoref{prop:almostregulardistr}, are explored in \\autoref{sec:liftingalmostreg}.\n\\begin{rem} \\autoref{prop:regularpipoissonlift} is false when $\\mathcal{F}$ is no longer regular, as the lifting property cannot be tested purely by looking at the underlying singular foliation of the Lie algebroid. For example, on $\\mathbb{R}^2$ the Lie algebroids $\\mathcal{A}_Z^k = \\langle x^k \\partial_x, \\partial_y \\rangle$ associated to the jets of $x \\in C^\\infty(\\mathbb{R}^2)$ for $Z = \\{x = 0\\}$ all have the same orbits, but $\\pi = x \\partial_x \\wedge \\partial_y$ only admits $\\mathcal{A}_Z^k$-lifts for $k = 0$ or $1$. Essentially what is used in the regular case is that a decomposition into leaves determines a unique module of vector fields if the decomposition is regular (see the discussion in \\cite{AndroulidakisZambon16}).\n\\end{rem}\n\\subsection{Lifting Poisson structures of divisor-type}\n\\label{sec:liftingdivtype}\nIn this section we discuss the process of lifting Poisson structures of divisor-type. Let $I \\subseteq C^\\infty(X)$ be a divisor ideal, and assume that the associated ideal Lie algebroid $TX_I \\to X$ exists. In other words, assume that the divisor ideal $I$ is projective. Recall that given a Lie algebroid $\\mathcal{A}$ we denote the spaces of $\\mathcal{A}$-liftable and $I$-divisor-bivectors by $\\mathfrak{X}^2_{\\mathcal{A}}(X)$ and $\\mathfrak{X}^2_{I}(X)$ respectively. Similarly, we have their Poisson counterparts ${\\rm Poiss}_{\\mathcal{A}}(X) \\subseteq \\mathfrak{X}^2_{\\mathcal{A}}(X)$ and ${\\rm Poiss}_{I}(X) \\subseteq \\mathfrak{X}^2_I(X)$.\n\nWe wish to study the relation between the spaces ${\\rm Poiss}_{TX_I}(X)$ and ${\\rm Poiss}_{I}(X)$. Note that for $\\pi \\in {\\rm Poiss}_I(X)$ to be liftable to $\\mathcal{A}$, by \\autoref{prop:apoissondivtypelifts} $I$ must be $I$ divisible by $I_\\mathcal{A}$. Consequently whether $I$ is standard (\\autoref{defn:divprojstandard}) is relevant, i.e.\\ whether $I_{TX_I} = I$.\n\nDivisor bivectors are generally not liftable to their ideal Lie algebroid, i.e.\\ $\\mathfrak{X}^2_I(X) \\not\\subseteq \\mathfrak{X}^2_{TX_I}(X)$. The other inclusion is also false, $\\mathfrak{X}^2_{TX_I}(X) \\not\\subseteq \\mathfrak{X}^2_I(X)$, as the following examples show.\n\\begin{exa}\\label{exa:logbivector} The bivector $\\pi = x \\partial_x \\wedge \\partial_y + \\partial_z \\wedge \\partial_w + \\partial_x \\wedge \\partial_w$ on $\\mathbb{R}^4$ with coordinates $(x,y,z,w)$ satisfies $\\wedge^2 \\pi = x \\partial_x \\wedge \\partial_y \\wedge \\partial_z \\wedge \\partial_w$, which vanishes transversally on $Z_\\pi = \\{x = 0\\}$. Hence $\\pi$ is of log divisor-type, but does not lift to $\\mathcal{A}_{Z_\\pi} = \\langle x \\partial_x, \\partial_y, \\partial_z, \\partial_w \\rangle$ due to the presence of $\\partial_x \\wedge \\partial_w$, showing that $\\mathfrak{X}^2_I(X) \\not\\subseteq \\mathfrak{X}^2_{TX_I}(X)$ in this case. Note that $\\pi$ is not Poisson, as $[\\pi,\\pi] = -\\partial_x \\wedge \\partial_y \\wedge \\partial_w \\neq 0$.\n\\end{exa}\n\\begin{exa} The Poisson bivector $\\pi = x^2 \\partial_x \\wedge \\partial_y$ on $\\mathbb{R}^2$ with coordinates $(x,y)$ lifts to $\\mathcal{A}_Z$, where $Z = \\{x = 0\\}$, with $\\mathcal{A}_Z$-lift $\\pi_{\\mathcal{A}_Z} = x (x \\partial_x) \\wedge \\partial_y$, but it does not lift nondegenerately, nor does $\\pi$ specify a log divisor structure on $Z$. Thus we see that $\\mathfrak{X}^2_{TX_I}(X) \\not\\subseteq \\mathfrak{X}^2_I(X)$ in this case.\n\\end{exa}\nThe Poisson condition is important for liftability as the above examples show. To study this in more detail, we first discuss which vector fields are liftable to $TX_I$.\n\\begin{prop}\\label{prop:liftifpreserve} Let $\\pi \\in \\mathfrak{X}^2_I(X)$. A vector field $V \\in \\mathfrak{X}(X)$ lies in $\\mathfrak{X}_{TX_I}(X)$ if $\\mathcal{L}_V (\\wedge^n \\pi) = 0$.\n\\end{prop}\n\\bp Denote $\\wedge^n \\pi$ by $\\pi^n$. By definition of $\\Gamma(TX_I) = \\Gamma(TX)_I$, we must show that $\\mathcal{L}_V I \\subset I$, which is a local statement. Let ${\\rm vol}_X$ be a local volume form and define $f := \\pi^n({\\rm vol}_X)$ so that $I = \\langle f \\rangle$. Then $\\mathcal{L}_V f = \\mathcal{L}_V \\pi^n({\\rm vol}_X) = (\\mathcal{L}_V \\pi^n)({\\rm vol}_X) + \\pi^n \\mathcal{L}_V \\rm{vol}_X$. Note that $\\mathcal{L}_V {\\rm vol}_X = g {\\rm vol}_X$ for some local function $g$. From this we conclude that\n\\begin{equation*}\n\t\\mathcal{L}_V f= \\mathcal{L}_V \\pi^n ({\\rm vol}_X) + \\pi^n(g {\\rm vol}_X) = \\mathcal{L}_V \\pi^n ({\\rm vol}_X) + g \\pi^n({\\rm vol}_X) = \\mathcal{L}_V \\pi^n ({\\rm vol}_X) + g f.\n\\end{equation*}\nAs $g f \\in I$, we see that $V$ lifts if $\\mathcal{L}_V \\pi^n = 0$.\n\\end{proof}\nThe assumption in \\autoref{prop:liftifpreserve} seems strong, yet allows for the following conclusion.\n\\begin{cor}\\label{cor:poissonhamiltonian} Let $\\pi \\in {\\rm Poiss}_I(X)$. Then ${\\rm Ham}_\\pi(X) \\subseteq {\\rm Poiss}_{\\pi}(X) \\subseteq \\mathfrak{X}_{TX_I}(X)$.\n\\end{cor}\n\\bp That ${\\rm Ham}_\\pi(X) \\subseteq {\\rm Poiss}_{\\pi}(X)$ is clear. Given $V \\in {\\rm Poiss}_\\pi(X)$ we have $\\mathcal{L}_{V} \\pi = 0$. But then we compute that $\\mathcal{L}_{V} \\pi^n = n (\\mathcal{L}_V \\pi) \\wedge \\pi^{n-1} = 0$, so that $V \\in \\mathfrak{X}_{TX_I}(X)$ by \\autoref{prop:liftifpreserve}.\n\\end{proof}\n\\begin{rem} For a general bivector $\\pi \\in \\mathfrak{X}^2(X)$, the definition of its `Hamiltonian vector fields' $V_f = \\pi^\\sharp(df)$ for $f \\in C^\\infty(X)$ still makes sense. However, the fact that all such vector fields preserve $\\pi$, i.e.\\ $\\mathcal{L}_{V_f} \\pi = 0$, is equivalent to $\\pi$ being Poisson.\n\\end{rem}\nDue to \\autoref{cor:poissonhamiltonian}, we see that for $\\pi \\in {\\rm Poiss}_I(X)$ there exists a map $\\widetilde{\\pi}^\\sharp\\colon T^*X \\to TX_I$ fitting in the following diagram, which is not yet the existence of an $TX_I$-lift.\n\\begin{center}\n\t\\begin{tikzpicture}\n\t%\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\tTX^*_I & TX_I \\\\ T^*X & TX \\\\};\n\t\\path[-stealth]\n\t(m-2-1) edge node [left] {$\\rho_{TX_I}^*$} (m-1-1)\n\t(m-2-1) edge node [above] {$\\pi^\\sharp$} (m-2-2)\n\t(m-2-1) edge node [above] {$\\widetilde{\\pi}^\\sharp$} (m-1-2)\n\t(m-1-2) edge node [right] {$\\rho_{TX_I}$} (m-2-2);\n\t\\draw[dotted, ->] (m-1-1) to (m-1-2);\n\t\\end{tikzpicture}\n\\end{center}\nOne can obtain an actual $TX_I$-lift under the assumption that $\\Gamma(TX_I^*)$ admits local bases of closed sections. Namely, we can now prove \\autoref{thm:introlifting} from the introduction.\n\\begin{thm}\\label{thm:ipoissonaitype} Let $I \\subseteq C^\\infty(X)$ be a projective divisor ideal with $TX^*_I$ admitting local bases of closed sections. Then ${\\rm Poiss}_I(X) \\subseteq {\\rm Poiss}_{TX_I}(X)$, i.e.\\ those of $I$-divisor-type are of $TX_I$-type.\n\\end{thm}\n\\bp We obtain from the discussion above a map $\\widetilde{\\pi}^\\sharp\\colon T^*X \\to TX_I$. We can dualize the bundle morphism $\\widetilde{\\pi}^\\sharp$ to a map $(\\widetilde{\\pi}^\\sharp)^*\\colon TX_I^* \\to TX$. To test whether this lifts to a map $\\pi_{TX_I}^\\sharp\\colon TX_I^* \\to TX_I$, we proceed as in \\autoref{cor:poissonhamiltonian} by checking whether $(\\widetilde{\\pi}^\\sharp)^*$ maps to $\\Gamma(TX)_I \\cong \\Gamma(TX_I)$ on sections. Assuming that $\\Gamma(TX_I^*)$ admits local bases of closed sections, we can by continuity answer this in the isomorphism locus $X \\backslash Z_I$, where such local sections are exact under the isomorphism with $TX$, and where $(\\widetilde{\\pi}^\\sharp)^* = (\\pi^\\sharp)^*$ using the isomorphism given by $\\rho_{TX_I}$. Here the lifting property follows as in \\autoref{cor:poissonhamiltonian}, using that $(\\pi^\\sharp)^* = - \\pi^\\sharp$ by skew-symmetry. The lifting property then holds in the entirety of $X$ by density of the isomorphism locus.\n\\end{proof}\n\\begin{rem} By inspecting the proof of \\autoref{thm:ipoissonaitype}, we readily see that it still holds for Poisson structures of $m$-$I$-divisor-type, that is, any almost-regular Poisson structure.\n\\end{rem}\nThe converse to the above theorem is false unless $\\pi \\in {\\rm Poiss}(X)$ is of nondegenerate $TX_I$-type and $I$ is standard (see the discussion below \\autoref{prop:apoissondivtypelifts}). Further, we know that if $I$ is standard and $\\pi \\in {\\rm Poiss}_I(X)$ is of $TX_I$-type, then $\\pi$ is in fact of nondegenerate $TX_I$-type.\n\\begin{rem} Following up on \\autoref{rem:nonstandard}, let $I \\subseteq C^\\infty(X)$ be a nontrivial projective divisor ideal and $\\pi \\in {\\rm Poiss}_{I^k}(X)$ for some $k > 1$. While $\\pi$ can be of $TX_I$-type using \\autoref{thm:ipoissonaitype}, it can never be of nondegenerate $TX_I$-type by comparing the divisor ideals of $\\pi$ and $TX_I$.\n\\end{rem}\nNote that if $\\pi$ is of $TX_I$-type, the degeneracy locus $Z_I$ must be a $\\pi$-Poisson subset, i.e.\\ $I_{Z_I}$ is a $\\pi$-Poisson ideal. Similarly we have the following (recall that $I \\subseteq I_{Z_I}$ by \\autoref{prop:divvanishingideal}).\n\\begin{prop}\\label{prop:itypeipoissonideal} Let $I$ be a divisor ideal and $\\pi \\in {\\rm Poiss}_I(X)$. Then $I$ is a $\\pi$-Poisson ideal.\n\\end{prop}\n\\bp Let ${\\rm vol}_X$ be a local volume form, and $g \\in C^\\infty(X)$. Then $I = \\langle \\pi^n({\\rm vol}_X) \\rangle = \\langle f \\rangle$, and\n\\begin{equation*}\n\t\\{g, f\\}_\\pi = \\{g, \\pi^n({\\rm vol}_X)\\}_\\pi = \\pi(dg, d(\\pi^n{\\rm vol}_X)) = \\pm \\mathcal{L}_{V_g} (d \\pi^n({\\rm vol}_X)) = \\pi^n \\mathcal{L}_{V_g} {\\rm vol}_X,\n\\end{equation*}\nusing that as $V_g$ is Hamiltonian, it preserves $\\pi^n$. We have $\\mathcal{L}_{V_g} {\\rm vol}_X = h {\\rm vol}_X$ for some function $h \\in C^\\infty(X)$, so that $\\pi^n \\mathcal{L}_{V_g} {\\rm vol}_X = h f \\in I$. This implies that $\\{g,f\\}_\\pi \\in I$ as desired.\n\\end{proof}\n\\subsection{Examples}\nIn this section we discuss several interesting examples of Poisson structures of divisor-type, and of the lifting procedure to their associated ideal Lie algebroids. We will only do so here for Poisson structures in the usual sense, as this is the main motivation for the development of the theory. However, it is quite possible, and certainly interesting, to consider $\\mathcal{A}$-Poisson structures of divisor-type for given, well-understood, Lie algebroids $\\mathcal{A} \\to X$, in cases where it is difficult to intrinsically capture when a Poisson structure is of $\\mathcal{A}$-type. Recall the examples of divisors and ideal Lie algebroids given in \\autoref{sec:divexamples} and \\autoref{exa:ideallalgebroids}.\n\nThroughout, this section let $X$ be a given $2n$-dimensional manifold.\n\\begin{exa}[Nondegenerate] Let $\\pi \\in {\\rm Poiss}(X)$ be a nondegenerate Poisson structure. Then $\\pi$ is of $C^\\infty(X)$-divisor-type, i.e.\\ $I_\\pi$ is the trivial divisor ideal. By \\autoref{prop:liftingnondegiso}, $\\pi$ is only of $\\mathcal{A}$-type for the Lie algebroid $\\mathcal{A} = TX$, and certainly lifts to it nondegenerately.\n\\end{exa}\n\\begin{exa}[Log-Poisson, \\cite{GuilleminMirandaPires14}]\\label{exa:logpoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ be a \\emph{log-Poisson structure}, i.e.\\ such that it is of log divisor-type. Then $I_\\pi = I_Z$ is a log divisor ideal. Regarding liftability of $\\pi$, recall from \\autoref{exa:zvx} that the log-tangent bundle $\\mathcal{A}_Z$ is the (primary) ideal Lie algebroid of $I_Z$. By \\autoref{thm:ipoissonaitype} we see that $\\pi$ is of $\\mathcal{A}_Z$-type. As $I_Z$ is standard, it is in fact of nondegenerate $\\mathcal{A}_Z$-type, so that it is dual to a \\emph{log-symplectic structure} \\cite{GuilleminMirandaPires14,GualtieriLi14,Cavalcanti17,MarcutOsornoTorres14}.\n\\end{exa}\nWe can determine the lifting property to log-tangent bundles $\\mathcal{A}_Z \\to X$ more generally.\n\\begin{prop}\\label{prop:azliftpoisson} Let $\\pi \\in {\\rm Poiss}(X)$ be given and let $Z \\subseteq X$ be a hypersurface. Then the following are equivalent:\n\\begin{itemize}\n\t\\item $Z$ is a $\\pi$-Poisson submanifold;\n\t\\item $I_Z$ is a $\\pi$-Poisson ideal;\n\t\\item $\\pi$ of $\\mathcal{A}_Z$-type.\n\\end{itemize}\n\\end{prop}\n\\bp This is a rephrasing of \\cite[Proposition 4.4.1]{Pym13}. The equivalence of the first two points is standard, as $Z$ is a closed submanifold. If $\\pi$ of $\\mathcal{A}_Z$-type then $Z$ is $\\pi$-Poisson, because $Z$ is the degeneracy locus of $\\mathcal{A}_Z$ (c.f.\\ \\autoref{prop:pidegenpipoisson}), or because it is an orbit of $\\mathcal{A}_Z$ (c.f.\\ \\autoref{prop:atypepoissonsubmfd}). To see that if $Z$ is $\\pi$-Poisson, then $\\pi$ is of $\\mathcal{A}_Z$-type, we turn to the proof of \\autoref{thm:ipoissonaitype}: there we showed first that each $\\pi$-Hamiltonian vector field is liftable to $\\mathcal{A}_Z$, before dualizing to obtain liftability of $\\pi$. However, $Z$ being $\\pi$-Poisson is equivalent to all $\\pi$-Hamiltonian vector fields being tangent to $Z$ (or preserving the vanishing ideal $I_Z$).\n\\end{proof}\nFrom \\autoref{prop:azliftpoisson} one can wonder when a Poisson structure $\\pi$ is of nondegenerate $\\mathcal{A}_Z$-type. However, this is already answered by \\autoref{prop:apoissondivtypelifts}: for this $\\pi$ must be of $I_Z$-divisor-type (because $I_{\\mathcal{A}_Z} = I_Z$), which is also sufficient by, for example, \\autoref{prop:itypeipoissonideal}. Hence, $\\pi$ is of nondegenerate $\\mathcal{A}_Z$-type if and only if it of $I_Z$-divisor-type, i.e.\\ is log-Poisson.\n\\begin{rem} Because $I_Z$ is a locally principal (divisor) ideal, \\autoref{prop:azliftpoisson} says that Poisson structures of $I$-divisor-type are liftable to $\\mathcal{A}_Z = TX_{I_Z}$ if $I_Z$ divides $I$. The converse statement is the content of \\autoref{prop:apoissondivtypelifts}. It would be interesting to determine whether there are other Lie algebroids of divisor-type for which a similar statement is true, or to show that this only happens in this case. We seem to crucially use that $I_Z$ is not only a divisor ideal, but in fact the vanishing ideal of its support. This only happens in the log setting.\n\\end{rem}\nWe can determine the lifting property for log-bivectors following \\cite[Lemma 138]{Frejlich11}.\n\\begin{prop}[\\cite{Frejlich11}]\\label{prop:liftazbivectors} Let $\\pi \\in \\mathfrak{X}^2_{I_Z}(X)$ be a bivector of log-divisor-type. Then $\\pi$ is of $\\mathcal{A}_Z$-type if and only if $\\pi^\\sharp(T^*_x X) \\subseteq T_x Z$ for some $x \\in Z_i$ for each connected component $Z_i \\subseteq Z$. \n\\end{prop}\n\\bp Given $x \\in Z$ with local generator $z$ of $I_Z$ we have $\\pi = \\partial_z \\wedge \\pi^\\sharp(dz) + \\nu$ with $\\nu^\\sharp(dz) \\equiv 0$. Then $\\wedge^n \\pi \\sim \\partial_z \\wedge \\pi^\\sharp(dz) \\wedge \\nu^{n-1}$. This decomposition also shows that for all $x \\in Z$, the condition $\\pi^\\sharp(T^*_x X) \\subseteq T_x Z$ is equivalent to the condition $\\pi^\\sharp(d_x I_Z) = 0$. If $\\pi^\\sharp(d_x I_Z) = 0$, then $\\pi^\\sharp(dz)|_Z = 0$, so that in turn $\\pi^\\sharp(dz) = x V$ for some $V \\in \\Gamma(TX)$. Thus $\\pi^\\sharp(d\\log z) = V$ which shows that $\\pi$ is of $\\mathcal{A}_Z$-type (pointwise, that $\\pi_x \\in \\wedge^2 \\mathcal{A}_{Z,x}$). The converse statement is immediate, because $Z$ is an $\\mathcal{A}_Z$-invariant submanifold. Next, the space $\\mathcal{Z}(\\pi) = \\{x \\in Z \\, : \\, \\pi^\\sharp(T^*_x x) \\subseteq T_x Z\\} \\subseteq Z$ is both open and closed, where for openness we use that if $x \\in \\mathcal{Z}(\\pi)$, then $\\pi^\\sharp(d_x z) = 0$, whence $\\nu_x^{n-1} \\neq 0$ by transversality. But then there exists some open $U$ around $x$ on which $\\nu^{n-1}$ is nonvanishing. As $\\wedge^n \\pi$ vanishes on $U \\cap Z$, this implies that $\\pi^\\sharp(dz)$ divides $\\nu^{n-1}$ on $U \\cap Z$. To not contradict transversality of $\\wedge^n \\pi$, we must have that $\\pi^\\sharp(dz) = 0$ on $U \\cap Z$. This in turn implies that $U \\cap Z \\subseteq \\mathcal{Z}(\\pi)$, which shows openness. The rest follows immediately.\n\\end{proof}\nEssentially, the above proposition uses the fact that transversality is an open condition.\n\\begin{rem}\\label{rem:examplelogbivectorlift} Comparing this to \\autoref{exa:logbivector}, we see that there $\\pi^\\sharp(dw) = \\partial_z + \\partial_x \\not\\in TZ$, and indeed the bivector $\\pi$ of that example does not lift to $\\mathcal{A}_Z$, as said by \\autoref{prop:liftazbivectors}.\n\\end{rem}\n\\begin{exa}[Normal-crossing log-Poisson] Let $\\pi\\in {\\rm Poiss}(X)$ be a \\emph{normal-crossing log-Poisson structure}, i.e.\\ such that it is of normal-crossing log divisor-type, with divisor ideal $I_{\\underline{Z}}$. The above discussion (that is, \\autoref{prop:azliftpoisson}) also holds for normal-crossing log-Poisson structures: a Poisson structure is of $\\mathcal{A}_{\\underline{Z}}$-type if and only if $I_{\\underline{Z}}$ is a $\\pi$-Poisson ideal, which is true in this case by \\autoref{prop:itypeipoissonideal}. Being of nondegenerate $\\mathcal{A}_{\\underline{Z}}$-type is again equivalent to being of $I_{\\underline{Z}}$-divisor-type. See also \\cite[Proposition 4.4.1; Proposition 4.4.2]{Pym13}, and \\cite{Klaasse17}.\n\\end{exa}\n\\begin{exa}[$b^k$-Poisson, \\cite{Scott16}]\\label{exa:bkpoissonlift} Another interesting class is given by the \\emph{$b^k$-Poisson structures} $\\pi_k \\in {\\rm Poiss}(X)$, which are of $I^k_Z$-divisor-type for $k \\geq 0$, where $I_Z$ is a log divisor ideal. This does not characterize $b^k$-Poisson structures. To do so, demand that $\\{\\cdot,\\cdot\\}_{\\pi_k}$ satisfies\n\\begin{equation*}\n\t\\{\\cdot,\\cdot\\}_{\\pi_k}\\colon I_Z \\times C^\\infty(X) \\to I_Z^k.\n\\end{equation*}\nIf this holds, then $\\wedge^n \\pi_k$ provides a local generator of $I_Z^k$, and its $(k-1)$-jet can be used to define the $b^k$-tangent bundle $\\mathcal{A}_Z^k$ as in \\autoref{exa:bktangentbundle}. It then follows that $\\pi_k$ is of nondegenerate $\\mathcal{A}_Z^k$-type, or of nondegenerate \\emph{$b^k$-type}. Consequently they are dual to $b^k$-symplectic structures. Note here that $b^1$-Poisson structures are just the log-Poisson structures of \\autoref{exa:logpoissonlift}.\n\\end{exa}\n\\begin{rem} Our notion of nondegenerate $b^k$-type is almost the same as that of $b^k$-type in \\cite{Scott16}. There is one difference, namely that there is a class of (non-canonically) isomorphic Lie algebroids, namely $\\mathcal{A}_Z^k$ with respect to different jet data, and Scott's $b^k$-type demands liftability to any one of these Lie algebroids. We instead extract the jet data from $\\pi_k$ itself.\n\\end{rem}\n\\begin{exa}[Scattering Poisson, \\cite{Lanius16}]\\label{exa:scatteringpoissonlift} Let $(X,Z)$ be a log pair. Then $\\pi \\in {\\rm Poiss}(X)$ is a \\emph{scattering Poisson structure} if its associated Poisson bracket satisfies\n\t%\n\t\\begin{equation*}\n\t\\{\\cdot,\\cdot\\}_\\pi\\colon C^\\infty(X) \\times C^\\infty(X) \\to I_Z, \\qquad \\text{and} \n\t\\qquad \\{\\cdot,\\cdot\\}_\\pi\\colon I_Z \\times C^\\infty(X) \\to I_Z^2.\n\t\\end{equation*}\n\t%\n\tThen $\\pi$ is of nondegenerate $\\mathcal{C}_Z$-type, and also of $\\mathcal{A}_Z$-type (because $\\mathcal{C}_Z$ is an $\\mathcal{A}_Z$-Lie algebroid).\n\\end{exa}\n\\begin{rem} A Poisson structure $\\pi \\in {\\rm Poiss}(X)$ is of $\\mathcal{C}_Z$-type if $Z$ is $\\pi$-Poisson, and the first jet of its $\\mathcal{A}_Z$-lift $\\pi_{\\mathcal{A}_Z}$ vanishes at $Z$ (using \\autoref{prop:azliftpoisson}). Moreover, $\\pi$ is of nondegenerate $\\mathcal{C}_Z$-type if and only if $Z = X_{\\pi_{\\mathcal{A}_Z}, 2n-2}$ as $\\pi_{\\mathcal{A}_Z}$-Poisson divisors (c.f.\\ \\cite[Lemma 5.2]{Lanius16}), which provides an alternative definition of $\\pi$ being scattering Poisson for the log pair $(X,Z)$.\n\\end{rem}\n\\begin{exa}[Elliptic Poisson, \\cite{CavalcantiGualtieri18}]\\label{exa:ellpoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ be an \\emph{elliptic Poisson structure}, i.e.\\ such that it is of elliptic divisor-type. Then $I_\\pi = I_{|D|}$ for an elliptic pair $(X,|D|)$. The (primary) ideal Lie algebroid of $I_{|D|}$ is the elliptic tangent bundle $\\mathcal{A}_{|D|}$ (see \\autoref{exa:ellvx}). By \\autoref{thm:ipoissonaitype}, the bivector $\\pi$ is of nondegenerate $\\mathcal{A}_{|D|}$-type because $I_{|D|}$ is standard. This recovers \\cite[Lemma 3.4]{CavalcantiGualtieri18}. More discussion of these structures can be found in \\cite{KlaasseLanius18, KlaasseLi18}, determining their Darboux models, Poisson cohomology and adjoint symplectic groupoids.\n\\end{exa}\n\\begin{exa}[Elliptic-log Poisson]\\label{exa:elllogpoissonlift} Let $\\pi \\in {\\rm Poiss}(X)$ be an \\emph{elliptic-log Poisson structure}, i.e.\\ such that it is of elliptic-log divisor-type. Then $I_\\pi = I_W$ for an elliptic-log ideal $I_W = I_Z \\cdot I_{|D|}$. The degeneracy locus of $\\pi$ is then given by $Z$, with $D$ being a $\\pi$-symplectic leaf. Consequently, by \\autoref{prop:azliftpoisson} we know that $\\pi$ is of $\\mathcal{A}_Z$-type. Naturally one wonders whether $\\pi$ is in fact of $\\mathcal{A}_W$-type (noting that $\\mathcal{A}_W = TX_{I_W}$ is an $\\mathcal{A}_Z$-Lie algebroid, c.f.\\ \\autoref{exa:ideallalgebroids}. However, this cannot be immediately answered by \\autoref{thm:ipoissonaitype} because $\\mathcal{A}_W^*$ does not admit local closed generators. However, the first part of the lifting procedure (the map $\\widetilde{\\pi}^\\sharp$) does not use this, so that certainly $\\pi$ lifts to a map $\\widetilde{\\pi}^\\sharp\\colon T^*X \\to \\mathcal{A}_W$. As $I_W$ is standard, we further know that if $\\pi$ is of $\\mathcal{A}_W$-type, then it is of nondegenerate $\\mathcal{A}_W$-type. These types of Poisson structures are studied in more detail in \\cite{KlaasseLanius18}, where in particular their Poisson cohomology is determined.\n\\end{exa}\nIt is very interesting to extend the above list of examples using a new class of divisor ideals.\n\\subsection{Lifting Poisson structures of $m$-divisor-type}\n\\label{sec:liftingalmostreg}\nWe next turn to lifting Poisson structures of $m$-$I$-divisor-type. To this end, let $\\pi \\in {\\rm Poiss}(X)$ be of $m$-divisor-type. Then $\\pi$ is almost-regular by \\autoref{cor:divtypealmostreg}, so that by \\autoref{prop:almostregulardistr} it has a unique associated involutive regular distribution $D_\\pi \\subseteq TX$ of $\\pi$-Poisson submanifolds. We can readily view this distribution as an injective Lie algebroid $\\mathcal{D}_\\pi$, and can wonder whether $\\pi$ can be lifted to it.\n\\begin{prop}\\label{prop:mdivtypelift} Let $\\pi \\in {\\rm Poiss}(X)$ be of $m$-divisor-type with its associated injective Lie algebroid $\\mathcal{D}_\\pi \\to X$. Then $\\pi$ is of $I_\\pi$-$\\mathcal{D}_\\pi$-type, where $I_\\pi$ is the divisor ideal of $\\pi$.\n\\end{prop}\n\\bp As discussed in \\autoref{sec:almostregularpoisson}, the regular distribution $D_\\pi$ obtained from $\\pi$ consists of $\\pi$-Poisson submanifolds. \\autoref{prop:regularpipoissonlift} thus immediately implies that $\\pi$ is of $\\mathcal{D}_\\pi$-type. If we consider the definition of $\\pi$, we note that its associated divisor $(K, \\wedge^m \\pi)$ for $K \\subseteq \\wedge^{2m} TX$ is such that $\\det D_\\pi = K$. From this we see that its $\\mathcal{D}_\\pi$-lift is of $I_\\pi$-divisor-type as desired.\n\\end{proof}\nThis result is noticed without the above precise language in \\cite[Section 4.2]{AndroulidakisZambon17}. Note that the $\\mathcal{D}_\\pi$-lift $\\pi_{\\mathcal{D}} \\in {\\rm Poiss}(\\mathcal{D}_\\pi)$ is automatically generically-nondegenerate by \\autoref{prop:apoissondivtypelifts}.\n\\begin{rem} \\autoref{prop:mdivtypelift} allows for an alternative definition of almost-regularity (or of being of $m$-divisor-type). Namely, an almost-regular Poisson structure is uniquely determined by a regular involutive distribution $D$, and a generically-nondegenerate $\\mathcal{D}$-Poisson structure (with $\\mathcal{D}$ the injective Lie algebroid of $D$). Indeed, if $\\pi_{\\mathcal{D}} \\in {\\rm Poiss}(\\mathcal{D})$ is generically-nondegenerate, then $\\pi := \\rho_D(\\pi_{\\mathcal{D}})$ is almost-regular. \\autoref{prop:mdivtypelift} shows the converse.\n\\end{rem}\nThe above remark shows a useful viewpoint on almost-regular Poisson structures, or those of $m$-divisor-type. Namely, they are just Poisson structures of divisor-type, except using the injective Lie algebroid $\\mathcal{D}$ instead of $TX$. We further remark here that a Poisson structure being almost-regular is different from saying that it is generically regular ($X_{\\pi,{\\rm reg}} = X_{\\pi,{\\rm max}}$), as its degeneracy must further be governed by a divisor (instead of more involved degeneration).\n\nGiven an almost-regular Poisson structure $\\pi$ with its $\\mathcal{D}$-lift $\\pi_{D}$, in light of \\autoref{prop:mdivtypelift} and the discussion in \\autoref{sec:liftingdivtype}, one can try to lift the $\\mathcal{D}$-Poisson structure $\\pi_{\\mathcal{D}}$ further, to the ideal Lie algebroid $\\mathcal{D}_{I_\\pi} \\to \\mathcal{D} \\to TX$. We next consider this for log divisors.\n\\subsubsection{The case of $m$-log-Poisson structures}\nLet $\\pi \\in {\\rm Poiss}(X)$ be an \\emph{$m$-log-Poisson structure}, i.e.\\ such that it is of $m$-log-divisor-type. Such structures are called \\emph{log-f Poisson structure} in \\cite{AndroulidakisZambon17}, but we favor our name due to its consistency with other types of almost-regular Poisson structures. Our discussion here is partially parallel to \\cite[Section 3.4]{AndroulidakisZambon17}. By definition $I_\\pi$ is a log divisor ideal, and the degeneracy locus $Z_\\pi$ of $\\pi$ is a hypersurface. For ease of notation we set $Z := Z_\\pi$ so that $I_\\pi = I_Z$, and moreover let $\\mathcal{D} := \\mathcal{D}_\\pi$ be the associated injective Lie algebroid.\n\nBy \\autoref{prop:mdivtypelift} we know that $\\pi$ is of log-$\\mathcal{D}$-type. A natural question to ask is whether each leaf of $D$ is log-Poisson. However, this need not necessarily be the case.\n\\begin{prop}[{\\cite[Lemma 3.7]{AndroulidakisZambon17}}]\\label{prop:mlogpoissonleaflogpoisson} Let $(X,Z,\\mathcal{D},\\pi)$ be an $m$-log-Poisson manifold. Then a leaf $P$ of $D$ is log-Poisson if and only if for all $x \\in P \\cap Z$, we have that $D_x + T_x Z = T_x X$.\n\\end{prop}\nTo find a Lie algebroid to which $\\pi$ lifts nondegenerately, we can try to form the primary ideal Lie algebroid $\\mathcal{D}_{I_Z}$, which would be the $(TZ \\cap TD,Z)$-rescaling of $\\mathcal{D}$. However, it is not clear a priori that the involutive submodule $\\Gamma(\\mathcal{D})_{I_Z} \\subseteq \\Gamma(\\mathcal{D})$ is projective. For this, we have:\n\\begin{prop}\\label{prop:injectivelogalgebroid} Let $(X,Z)$ be a log pair and $\\mathcal{B} \\to X$ be an injective Lie algebroid. Then the $C^\\infty(X)$-module $\\Gamma(\\mathcal{B})_{I_Z}$ is projective if $Z$ is transverse to the orbits of $\\mathcal{B}$.\n\\end{prop}\n\\bp The orbits of $\\mathcal{B}$ are the leaves of the associated distribution $T\\mathcal{B} := \\rho_{\\mathcal{B}}(\\mathcal{B}) \\subseteq TX$. If $Z$ is transverse to this regular foliation, we can find local bases of generating vector fields $\\partial_{x_i}$ for $\\mathcal{B}$ and $\\partial_{y_j}$ for $TZ$ so that $\\langle \\partial_{x_1}, \\partial_{y_j} \\rangle = \\Gamma(TX)$. Then if locally $I_Z = \\langle z \\rangle$, we have that $\\mathcal{B}_{I_Z} = \\langle z \\partial_{z}, \\partial_{x_i} \\, : \\, i \\geq 2 \\rangle$, with $x_1 = z$, which is thus projective (note that $\\Gamma(TX)_{I_Z} = \\langle z \\partial_{z}, \\partial_{y_j} \\rangle$).\n\\end{proof}\n\\begin{rem}\\label{rem:dtangentZ} When $Z$ is everywhere tangent to the orbits of $\\mathcal{B}$, the module $\\Gamma(\\mathcal{B})_{I_Z}$ is also projective, but just equals $\\Gamma(\\mathcal{B})$. Thus in this case we have that $\\mathcal{B}_{I_Z} = \\mathcal{B}$, i.e.\\ $\\mathcal{B}$ is unchanged.\n\\end{rem}\nWe can apply \\autoref{prop:injectivelogalgebroid} to the case where $\\mathcal{B} = \\mathcal{D}$. This shows that if $Z$ is transverse to the orbits of $\\mathcal{D}$, then $\\mathcal{D}_{I_Z}$ is an almost-injective Lie algebroid for which the natural map $\\mathcal{D}_{I_Z} \\to \\mathcal{D}$ is of $I_Z$-divisor-type. Moreover it follows from the discussion in \\autoref{sec:liftingdivtype} that the $m$-log-Poisson structure $\\pi$ is not only of $\\mathcal{D}$-type, but also of $\\mathcal{D}_{I_Z}$-type. We conclude that when $Z$ is transverse to the orbits of $\\mathcal{D}$, then $\\pi$ is of nondegenerate $\\mathcal{D}_{I_Z}$-type by \\autoref{prop:apoissondivtypelifts}.\n\nAn alternate approach is to consider $\\pi$ and whether it lifts to the usual ideal Lie algebroid $TX_{I_Z} = \\mathcal{A}_Z$, the log-tangent bundle. As $Z$ is by definition the degeneracy locus of $\\pi$, it is a $\\pi$-Poisson submanifold by \\autoref{prop:pidegenpipoisson}. From this we conclude immediately using \\autoref{prop:azliftpoisson} that $\\pi$ is of $\\mathcal{A}_Z$-type with lift $\\pi_{\\mathcal{A}_Z}$. In fact, it follows that $\\pi_{\\mathcal{A}_Z}$ is $2m$-regular when $Z$ is transverse to the orbits of $\\mathcal{D}$. Similarly, when $Z$ is everywhere tangent to the orbits of $\\mathcal{D}$, then $\\pi_{\\mathcal{A}_Z}$ is of $m$-log-divisor-type. By construction $\\mathcal{D}_{I_Z}$ is a $\\mathcal{D}$-Lie algebroid, but from the proof of \\autoref{prop:injectivelogalgebroid} we see that $\\mathcal{D}_{I_Z}$ is also an $\\mathcal{A}_Z$-Lie algebroid (also \\autoref{rem:dtangentZ}). This summarizes into the following diagram.\n\\begin{center}\n\t\\begin{tikzpicture}\n\t%\n\t\\matrix (m) [matrix of math nodes, row sep=2.5em, column sep=2.5em,text height=1.5ex, text depth=0.25ex]\n\t{\t(\\mathcal{D}_{I_Z}, \\pi_{\\mathcal{D}_{I_Z}}) & (\\mathcal{D}, \\pi_{\\mathcal{D}}) \\\\ (\\mathcal{A}_Z, \\pi_{\\mathcal{A}_Z}) & (TX, \\pi) \\\\};\n\t\\path[-stealth]\n\t(m-2-1) edge node [above] {$\\rho_{\\mathcal{A}_Z}$} (m-2-2)\n\t(m-1-1) edge node [above] {$\\rho_{\\mathcal{A}_Z}$} (m-1-2);\n\t\\draw[right hook-latex]\n\t(m-1-1) edge (m-2-1)\n\t(m-1-2) edge (m-2-2);\n\t\\end{tikzpicture}\n\\end{center}\nThus, an $m$-log-Poisson structure $\\pi \\in {\\rm Poiss}(X)$ with divisor ideal $I_Z$ and distribution $D$:\n\\begin{itemize}\n\t\\item Is always of log-$\\mathcal{D}$-type;\n\t\\item Is always of $\\mathcal{A}_Z$-type, with $2m$-regular lift when $Z$ is transverse to the $\\mathcal{D}$ orbits;\n\t\\item Is of nondegenerate $\\mathcal{D}_{I_Z}$-type when $Z$ is transverse to the $\\mathcal{D}$-orbits.\n\\end{itemize}\nAt this point it is clear that it is important to understand the relation between transversality of $Z$ and the orbits of $\\mathcal{D}$, and Poisson geometric properties of $\\pi$. The answer to this is provided by the behavior of the modular foliation $\\mathcal{F}_{\\pi,{\\rm mod}}$ of $\\pi$, which in general satisfies $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}}$ (discussed in \\autoref{sec:modularfoliation}, see \\autoref{exa:modfolmlog} in particular). Following \\cite{AndroulidakisZambon17}, we denote by\n\\begin{equation*}\n\tZ_{\\rm sing} = \\{x \\in Z \\, | \\, D_x \\subseteq T_x Z\\}\n\\end{equation*}\nthe points on $Z$ where $Z$ does not intersect the orbits of $\\mathcal{D}$ transversely. Due to \\autoref{prop:mlogpoissonleaflogpoisson}, these are the points on $Z$ whose associated leaf $P$ of $D$ is not log-Poisson. The two extreme cases are where $Z_{\\rm sing} = \\emptyset$, so that $Z$ is everywhere transverse to the orbits of $\\mathcal{D}$, and the case $Z_{\\rm sing} = Z$, where $\\mathcal{D}$ is tangent to $Z$. The general setting is more complicated.\n\nWe next follow the above discussion for the three examples contained in \\cite[Example 3.8]{AndroulidakisZambon17}.\n\\begin{exa}\\label{rem:mlogpoissonlocal} Consider the following three $1$-log-Poisson structures on $(X =\\mathbb{R}^3,(x,y,z))$:\n\\begin{itemize}\n\t\\item $\\pi_1 = x \\partial_x \\wedge \\partial_y$;\n\t\\item $\\pi_2 = z \\partial_x \\wedge \\partial_y$;\n\t\\item $\\pi_3 = (z - x^2) \\partial_x \\wedge \\partial_y$.\n\\end{itemize}\nWe then have that $\\Gamma(TX) = \\langle \\partial_x, \\partial_y, \\partial_z \\rangle$ and moreover $\\mathcal{D} := \\mathcal{D}_1 = \\mathcal{D}_2 = \\mathcal{D}_3 = \\langle \\partial_x, \\partial_y \\rangle$. To proceed we list some of the relevant objects for each of these (where $I_i := I_{\\pi_i}$ for $i = 1,2,3$):\n\\begin{itemize}\n\t\\item $I_1 = \\langle x \\rangle$, $\\Gamma(TZ_1) = \\langle \\partial_y, \\partial_z \\rangle$, $Z_{1,{\\rm sing}} = \\emptyset$, $\\Gamma(\\mathcal{A}_{Z_1}) = \\langle x \\partial_x, \\partial_y, \\partial_z \\rangle$, $\\mathcal{D}_{I_1} = \\langle x \\partial_x, \\partial_y \\rangle$;\n\t\\item $I_2 = \\langle z \\rangle$, $\\Gamma(TZ_2) = \\langle \\partial_x, \\partial_y \\rangle$, $Z_{2,{\\rm sing}} = Z_2$, $\\Gamma(\\mathcal{A}_{Z_2})  = \\langle \\partial_x, \\partial_y, z \\partial_z \\rangle$, $\\mathcal{D}_{I_2} = \\langle \\partial_x, \\partial_y \\rangle = \\mathcal{D}$.\n\t\\item $I_3 = \\langle z - x^2 \\rangle$, $\\Gamma(TZ_3) = \\langle \\partial_x + 2x \\partial_z, \\partial_y \\rangle$, $Z_{3,{\\rm sing}} = \\{x_1 = 0\\} \\cap Z$.\n\t\\end{itemize}\nFor $\\pi_1$ and $\\pi_2$, $Z$ is either everywhere transverse, or everywhere tangent to the orbits of $\\mathcal{D}_i$. The case of $\\pi_3$ is more involved, as is the behavior of the modules $\\Gamma(TX)_{I_{Z_3}}$ and $\\Gamma(\\mathcal{D})_{I_{Z_3}}$.\n\nThis shows that $\\pi_i$ lifts to be of log-type to $\\mathcal{D}$ for $i = 1,2,3$, as in \\autoref{prop:mdivtypelift}. Moreover, the $\\mathcal{A}_{Z_i}$-lifts of the Poisson structures $\\pi_i$ for $i = 1,2$ are given by\n\\begin{equation*}\n\t\\pi_{\\mathcal{A}_{Z_1}} = (x \\partial_x) \\wedge \\partial_y, \\quad \\pi_{\\mathcal{A}_{Z_2}} = z \\partial_x \\wedge \\partial_y.\n\\end{equation*}\nThus $\\pi_{\\mathcal{A}_{Z_1}}$ is $2$-regular, $\\pi_{\\mathcal{A}_{Z_2}}$ is of $1$-log-divisor-type, and $\\pi_1$ is of nondegenerate $\\mathcal{D}_{I_1}$-type.\n\\end{exa}\n\\subsection{Symplectic Lie algebroids}\n\\label{sec:symplecticliealgebroids}\nIn this section we briefly discuss symplectic Lie algebroids \\cite{NestTsygan01}, the associated study of $\\mathcal{A}$-symplectic geometry, and its relation with $\\mathcal{A}$-Poisson geometry.\n\\begin{defn}[\\cite{NestTsygan01}] A \\emph{symplectic Lie algebroid} is a Lie algebroid $\\mathcal{A} \\to X$ equipped with an \\emph{$\\mathcal{A}$-symplectic structure}, i.e.\\ a closed nondegenerate $\\mathcal{A}$-two-form ${\\omega_{\\mathcal{A}} \\in \\Omega^2_{\\mathcal{A}}(X)}$.\n\\end{defn}\nIn other words, the $\\mathcal{A}$-two-form $\\omega_\\mathcal{A}$ satisfies $d_\\mathcal{A} \\omega_\\mathcal{A} = 0$ and $\\omega_\\mathcal{A}^n \\neq 0$, where ${\\rm rank}(\\mathcal{A}) = 2n$. The nondegeneracy condition for $\\omega_\\mathcal{A}$ is equivalent to the map $\\omega_\\mathcal{A}^\\flat\\colon \\mathcal{A}^* \\to \\mathcal{A}$, given by $v \\mapsto \\iota_v \\omega_\\mathcal{A}$ for $v \\in \\mathcal{A}$, being an isomorphism. The space of $\\mathcal{A}$-symplectic forms is denoted by ${\\rm Symp}(\\mathcal{A})$, and we say that $X$ is \\emph{$\\mathcal{A}$-symplectic} if it is equipped with a symplectic Lie algebroid $(\\mathcal{A},\\omega_{\\mathcal{A}}) \\to X$.\n\nThe nondegeneracy condition for $\\omega_{\\mathcal{A}}$ implies that the rank of $\\mathcal{A}$ must be even (but note that $\\dim(X)$ need not necessarily be even). Any symplectic Lie algebroid $(\\mathcal{A},\\omega_\\mathcal{A})$ defines an $\\mathcal{A}$-cohomology class $[\\omega_\\mathcal{A}] \\in H^2(\\mathcal{A})$, is naturally oriented by $\\omega_\\mathcal{A}^n \\in \\Gamma(\\det(\\mathcal{A}^*))$, and always admits an \\emph{$\\mathcal{A}$-almost-complex structure}, i.e.\\ a complex structure $J_\\mathcal{A} \\in {\\rm End}(\\mathcal{A})$ for $\\mathcal{A}$ (see \\cite{Klaasse18three}). Lie algebroids $\\mathcal{A}$ of rank two admit $\\mathcal{A}$-symplectic structures if and only if they are orientable, i.e.\\ if $w_1(\\mathcal{A}) = 0$.\nThere is a standard notion of morphism between $\\mathcal{A}$-symplectic manifolds.\n\\begin{defn} Let $(\\varphi,f)\\colon (X, \\mathcal{A}, \\omega_\\mathcal{A}) \\to (X', \\mathcal{A}', \\omega_{\\mathcal{A}'})$ be a Lie algebroid morphism between symplectic Lie algebroids. Then the map $(\\varphi,f)$ is\n\\begin{itemize}\n\t\\item \\emph{$\\mathcal{A}$-symplectic} if $\\varphi^* \\omega_{\\mathcal{A}'} = \\omega_\\mathcal{A}$;\n\t\\item an \\emph{$\\mathcal{A}$-symplectomorphism} if it $\\mathcal{A}$-symplectic, and $\\varphi$ is a fiberwise isomorphism.\n\\end{itemize}\n\\end{defn}\nThe $\\mathcal{A}$-symplectic condition $\\varphi^* \\omega_{\\mathcal{A}'} = \\omega_\\mathcal{A}$ is equivalent to demanding $\\omega_\\mathcal{A}^\\flat = \\varphi \\circ \\omega_\\mathcal{B}^\\flat \\circ \\varphi^*$ as maps. As both $\\omega_\\mathcal{A}$ and $\\omega_{\\mathcal{A}'}$ are nondegenerate, any $\\mathcal{A}$-symplectic map $(\\varphi,f)$ must in particular be fiberwise injective. Consequently, if ${\\rm rank}(\\mathcal{A}) = {\\rm rank}(\\mathcal{A}')$, any $\\mathcal{A}$-symplectic map is necessarily an $\\mathcal{A}$-symplectomorphism, but the base map $f\\colon X \\to X'$ is not necessarily a diffeomorphism. Our main interest however will be in Lie algebroid isomorphisms which are $\\mathcal{A}$-symplectic.\n\\begin{rem} If $\\mathcal{A} \\to X$ is a Lie algebroid of divisor-type, one can perform any symplectic operation to a given $\\mathcal{A}$-symplectic structure in the isomorphism locus $X_\\mathcal{A}$. These include for example the symplectic fiber sum \\cite{Gompf95} or symplectic blow-up \\cite{McDuffSalamon17} procedures.\n\\end{rem}\nLet $(\\mathcal{A},\\omega_{\\mathcal{A}}) \\to X$ be a symplectic Lie algebroid. Then the map $\\omega_{\\mathcal{A}}^\\flat\\colon \\mathcal{A} \\to \\mathcal{A}^*$ can be inverted to a map $(\\omega_{\\mathcal{A}}^\\flat)^{-1}\\colon \\mathcal{A}^* \\to \\mathcal{A}$, which we will denote by $\\pi_{\\mathcal{A}}^\\flat$. This construction defines an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}} \\in {\\rm Poiss}(\\mathcal{A})$, which we will also denote by $\\pi_{\\mathcal{A}} = \\omega_{\\mathcal{A}}^{-1}$. Indeed, the conditions $d\\omega_\\mathcal{A} = 0$ and $[\\pi_{\\mathcal{A}},\\pi_{\\mathcal{A}}]_\\mathcal{A} = 0$ are equivalent. As discussed below \\autoref{defn:apoissonmap}, the two-form $\\omega_{\\mathcal{A}}$ is called the \\emph{dual $\\mathcal{A}$-form} to the underlying Poisson structure $\\pi := \\rho_{A}(\\pi_{\\mathcal{A}})$.\n\nThe following is the Moser theorem for Lie algebroids $\\mathcal{A}$ of smooth divisor-type \\cite{KlaasseLanius18}. It covers several results in the literature (\\cite{GuilleminMirandaPires14,Lanius16,MarcutOsornoTorres14,MirandaPlanas18,Moser65,NestTsygan96,Radko02,Scott16}), and is similar to \\cite{MirandaScott18}.\n\\begin{thm}[\\cite{KlaasseLanius18}]\\label{thm:amoser} Let $\\mathcal{A} \\to X$ be a Lie algebroid of smooth divisor-type, and let $k \\in \\{1, 2, \\dim X\\}$. Let $\\omega, \\omega' \\in \\Omega^k(\\mathcal{A})$ be $d_\\mathcal{A}$-closed $\\mathcal{A}$-$k$-forms that are nondegenerate on $Z_\\mathcal{A}$. Assume that either\n\t%\n\t\\begin{itemize}\n\t\\item $[\\omega] = [\\omega'] \\in H^k(\\mathcal{A})$, or\n\t\\item $\\widetilde{\\omega} - \\omega = \\rho_{A}^* \\tau'$ for $\\tau' \\in \\Omega^k(X)$ satisfying $\\tau'|_{Z_\\mathcal{A}} = 0$.\n\t\\end{itemize}\n\t%\n\tThen there exists a Lie algebroid isomorphism $(\\varphi,f)\\colon (\\mathcal{A}|_{U},\\omega) \\to (\\mathcal{A}|_{U'},\\omega')$ on neighbourhoods $U$ and $U'$ of $Z_\\mathcal{A}$ for which $\\varphi^* \\omega' = \\omega$, which in the second case can be chosen such that $f|_{Z_\\mathcal{A}} = {\\rm id}$.\n\\end{thm}\n\\begin{rem} As a consequence of \\autoref{thm:amoser}, to establish an $\\mathcal{A}$-Darboux theorem providing a pointwise normal form for $\\mathcal{A}$-symplectic structures, one need only establish what an $\\mathcal{A}$-symplectic structure must look like locally at a point in $Z_\\mathcal{A}$.\n\\end{rem}\nThis result implies that $\\mathcal{A}$-Nambu structures of top degree (nonvanishing sections of $\\det(\\mathcal{A})$), and hence $\\mathcal{A}$-symplectic structures on surfaces, specifying the same $\\mathcal{A}$-orientation are classified by their $\\mathcal{A}$-cohomology class. Namely, letting $n = \\dim X$, we note that $\\det(\\mathcal{A}^*)$ is a line bundle. Consequently, given cohomologous forms $\\omega,\\omega' \\in \\Omega^n(\\mathcal{A}) = \\Gamma(\\det(\\mathcal{A}^*))$ we have $\\omega = f \\omega'$ for some nonvanishing function $f \\in C^\\infty(X)$. This function must be strictly positive as $\\omega$ and $\\omega'$ give rise to the same $\\mathcal{A}$-orientation, so that $(1-t)\\omega + t \\omega' = ((1-t) + t f) \\omega$ is nondegenerate for all $t \\in [0,1]$. With more knowledge about the Lie algebroid $\\mathcal{A}$, the assumption on induced $\\mathcal{A}$-orientations can sometimes be dropped. This discussion summarizes as follows.\n\\begin{prop} Let $\\mathcal{A} \\to X$ be a Lie algebroid of divisor-type for which $Z_\\mathcal{A}$ is smooth. An $\\mathcal{A}$-Nambu structure $\\Pi$ inducing a given $\\mathcal{A}$-orientation is classified up to $\\mathcal{A}$-orientation-preserving isomorphism by its $\\mathcal{A}$-cohomology class.\n\\end{prop}\nSee also \\cite{MartinezTorres04} and \\cite{MirandaPlanas18} for the cases of log- and $b^k$-Nambu structures of top degree. Let us reiterate this classifies such $\\mathcal{A}$-symplectic structures on surfaces using $H^2(\\mathcal{A})$. Below we give some examples of local Darboux theorems for Poisson structures of divisor-type.\n\\begin{exa}[Nondegenerate] Let $\\pi \\in {\\rm Poiss}(X^{2n})$ be nondegenerate. Then $\\pi$ is dual to a symplectic structure. In other words, it is of nondegenerate $TX$-type. The classical Darboux theorem (e.g.\\ proven using Moser methods) states that around any point in $X$ there are coordinates in which $\\pi = \\omega_0^{-1}$, with $\\omega_0 = \\sum_i dx_i \\wedge dy_i$ the standard symplectic structure in $\\mathbb{R}^{2n}$.\n\\end{exa}\n\\begin{exa}[Log-Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be log-Poisson, i.e.\\ of log divisor-type with divisor ideal $I_Z$. Then $\\pi$ is of nondegenerate $\\mathcal{A}_Z$-type by \\autoref{exa:logpoissonlift}. Either directly by applying Weinstein's splitting theorem, or by Moser methods for $\\mathcal{A}_Z$ (see \\cite{GuilleminMirandaPires14}), around points in $Z$ there are coordinates $(z,x_i)$ with $I_Z = \\langle z \\rangle$ in which $\\pi = z \\partial_z \\wedge \\partial_{x_2} + \\omega_0^{-1}$.\n\\end{exa}\nThe normal-crossing log case is more involved, as there can be local cohomology of the Lie algebroid $\\mathcal{A}_{\\underline{Z}}$ in contractible neighbourhoods. See the discussions found in \\cite{Lanius16two,Lanius17,MirandaScott18,Radko02two}.\n\\begin{exa}[$b^k$-Poisson]\\label{exa:bkpoissondarboux} Let $\\pi \\in {\\rm Poiss}(X)$ be $b^k$-Poisson as in \\autoref{exa:bkpoissonlift}. Then $\\pi$ is of nondegenerate $\\mathcal{A}_Z^k$-type for the associated $b^k$-tangent bundle. By Moser methods for $\\mathcal{A}_Z^k$ (see \\cite{GuilleminMirandaWeitsman17,Scott16}), around points in $Z$ there are coordinates $(z,x_i)$ with $z \\in j_{k-1}$ such that\n\\begin{equation*}\n\t\\pi = z^k \\partial_z \\wedge \\partial_{x_1} + \\omega_0^{-1}.\n\\end{equation*}\n\\end{exa}\n\\begin{exa}[Scattering Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be scattering Poisson, so that $\\pi$ is of nondegenerate $\\mathcal{C}_Z$-type (as in \\autoref{exa:scatteringpoissonlift}). By applying Moser methods for $\\mathcal{C}_Z$ (see \\cite{Lanius16}), around points in $Z$ there are coordinates $(z,x_i)$ with $I_Z = \\langle z \\rangle$ in which (for $\\alpha_0$ the standard contact structure, so that $d\\alpha_0 = \\omega_0$)\n\\begin{equation*}\n\t\\pi = z^3 \\partial_z \\wedge \\partial_{x_1} + z^2 \\partial_{z} \\wedge \\alpha_0^{-1} + z^2 \\omega_0^{-1}.\n\\end{equation*}\n\\end{exa}\n\\begin{exa}[Elliptic Poisson]\\label{exa:ellpoissondarboux} Let $\\pi \\in {\\rm Poiss}(X)$ be elliptic Poisson, i.e.\\ of elliptic divisor-type with divisor ideal $I_{|D|}$. Then $\\pi$ is of nondegenerate $\\mathcal{A}_{|D|}$-type by \\autoref{exa:ellpoissonlift}. The local (and global) behavior of $\\pi$ depends on the value(s) of its elliptic residue ${\\rm Res}_q(\\pi)$. This is a residue map associated to the elliptic tangent bundle in \\cite{CavalcantiGualtieri18}, which we recall in \\autoref{exa:elltangentresidues}. There are two qualitatively different local Darboux models for $\\pi$, depending on the vanishing of ${\\rm Res}_q(\\pi)$ (see \\cite{KlaasseLanius18}). Around points in $D$ there are coordinates $(r,\\theta,x_i)$ with $I_{|D|} = \\langle r^2 \\rangle$ with:\n\\begin{equation*}\n\t\\pi = \\begin{cases} r \\partial_r \\wedge \\partial_{x_1} + \\partial_{\\theta} \\wedge \\partial_{x_2} + \\omega_0^{-1} \\qquad &\\text{if } {\\rm Res}_q(\\pi) = 0,\\\\ \\lambda r \\partial_r \\wedge \\partial_\\theta + \\omega_0^{-1} \\qquad &\\text{if } {\\rm Res}_q(\\pi) = \\lambda \\neq 0.\n\\end{cases}\n\\end{equation*}\nThis recovers the classification results of such Poisson structures on surfaces which can be found in \\cite{Radko02two} (where ${\\rm Res}_q(\\pi) \\neq 0$ is forced by nondegeneracy). See also \\cite{KlaasseLi18}.\n\\end{exa}\nThe local behavior of elliptic-log Poisson structures (\\autoref{exa:elllogpoissonlift}) is described in \\cite{KlaasseLanius18}.\n\\section{Residue maps and the modular foliation}\n\\label{sec:residues}\nIn this section we discuss two distinct but related concepts. The first of these is the residue maps one can associate to a Lie algebroid $\\mathcal{A}$ if one is given an appropriate $\\mathcal{A}$-invariant submanifold (\\autoref{defn:ainvariantresidues}). After this we will discuss Lie algebroid modules and consequently the Lie algebroid cohomologies they give rise to. We then discuss when the residue maps, which are initially defined on the levels of Lie algebroid forms, descend to the level of cohomology. After this we proceed to discuss the Lie algebroid modules in the presence of Poisson structures, and in particular consider the modular foliation. We finish by relating the residue maps to symplectic Lie algebroids and the modular foliation in concrete examples.\n\\subsection{Residue maps}\n\\label{sec:residuemaps}\nIn this section we discuss residues of Lie algebroid forms. These are a way to extract relevant information of $\\mathcal{A}$-forms over $\\mathcal{A}$-invariant submanifolds where the restriction of $\\mathcal{A}$ is projective. We further discuss how such residues interact with Lie algebroids morphisms. Residues are used when dealing with geometric structures on $\\mathcal{A}$, in order to extract information along the submanifold (see e.g.\\ \\cite{CavalcantiGualtieri18, CavalcantiKlaasse18}). In \\cite{CavalcantiKlaasse18} the following is discussed for transitive $\\mathcal{A}$-invariant submanifolds, but here we expose a generalization allowing for projective ones, recovering the transitive case when we can take $\\mathcal{B} = TD$ below. As a special case it further covers the case when the restriction of $\\mathcal{A}$ is regular.\n\nLet $\\mathcal{A} \\to X$ be a Lie algebroid and let $D \\subseteq X$ be a projective $\\mathcal{A}$-invariant submanifold (as in \\autoref{defn:lalgebroidadjective}), with $\\mathcal{B} \\to D$ and $(\\rho_A|_D)(\\Gamma(\\mathcal{A}|_D)) \\cong \\Gamma(\\mathcal{B})$. We obtain a short exact sequence\n\\begin{equation*}\n0 \\to \\ker \\widetilde{\\rho}_\\mathcal{A}|_D \\to \\mathcal{A}|_D \\to \\mathcal{B} \\to 0,\n\\end{equation*}\nwhere the $\\widetilde{\\rho}_\\mathcal{A}|_D\\colon \\mathcal{A}|_D \\to \\mathcal{B}$ is the induced bundle map coming from the map on sections. Note that in the transitive case we have that $\\widetilde{\\rho}_{\\mathcal{A}}|_D = \\rho_{A}|_D$.\nWe see that $\\mathcal{A}|_D$ is an extension of $\\mathcal{B}$ by $\\ker \\widetilde{\\rho}_\\mathcal{A}|_D$, the latter being the germinal isotropy of \\autoref{sec:singularliealgebroids}. The case in which we are mainly interested in is when $D = X \\backslash X_\\mathcal{A} = Z_\\mathcal{A}$, the complement of the isomorphism locus of $\\mathcal{A}$ (although in general the restriction need not be transitive nor projective). This is an $\\mathcal{A}$-invariant submanifold of $X$ if it is smooth by \\autoref{prop:degenlocus}, as $X \\backslash X_\\mathcal{A} = X_{\\mathcal{A},n-1}$ where $\\dim X = {\\rm rank}(\\mathcal{A}) = n$ (see \\autoref{exa:denseisolocus}). Note further that $X$ itself is always $\\mathcal{A}$-invariant, as is sometimes useful (say, for regular Lie algebroids). Dualizing the above sequence, we obtain\n\\begin{equation*}\n0 \\to \\mathcal{B}^* \\to \\mathcal{A}^*|_D \\to (\\ker \\widetilde{\\rho}_\\mathcal{A}|_D)^* \\to 0.\t\n\\end{equation*}\nWe are now in the following more general situation: given a short exact sequence $\\mathcal{S}\\colon 0 \\to E \\to W \\to V \\to 0$ of vector spaces, there is an associated dual sequence $\\mathcal{S}^*\\colon 0 \\to V^* \\to W^* \\to E^* \\to 0$. For a given $k \\in \\mathbb{N}$, by taking $k$th exterior powers we obtain a filtration of spaces $\\mathcal{F}^i := \\{\\rho \\in \\wedge^k W^* \\, | \\, \\iota_x \\rho = 0$ for all $x \\in \\wedge^i E \\}$, for $i = 0,\\dots,k+1$. These spaces satisfy $\\mathcal{F}^0 = 0$, $\\mathcal{F}^1 = \\wedge^k V^*$, $\\mathcal{F}^i \\subset \\mathcal{F}^{i+1}$, and $\\mathcal{F}^{i+1} \/ \\mathcal{F}^i \\cong \\wedge^{k-i} V^* \\otimes \\wedge^i E^*$. Setting $\\ell := \\dim E$, we have $\\mathcal{F}^{\\ell + 1} = \\wedge^k W^*$. The \\emph{residue} of an element $\\rho \\in \\wedge^k W^*$ is defined as its equivalence class ${\\rm Res}(\\rho) = [\\rho] \\in \\mathcal{F}^{\\ell + 1} \/ \\mathcal{F}^{\\ell} \\cong \\wedge^{k-\\ell} V^* \\otimes \\wedge^\\ell E^*$. Upon a choice of trivialization of $\\wedge^\\ell E^*$, i.e.\\ a choice of volume element for $E$, one can view the residue ${\\rm Res}(\\rho)$ as an element of $\\wedge^{k-\\ell} V^*$.\n\nWe now define the top residue of a Lie algebroid along a projective $\\mathcal{A}$-invariant submanifold, by performing the above operation to a smoothly varying section of $\\wedge^\\bullet \\mathcal{A}^*$ restricted to $D$.\n\\begin{defn}\\label{defn:ainvariantresidues} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $D \\subseteq X$ a projective $\\mathcal{A}$-invariant submanifold with image $\\mathcal{B}$ and germinal isotropy $E := \\ker \\widetilde{\\rho}_\\mathcal{A}|_D$. The \\emph{residue map} of $\\mathcal{A}$ along $D$ is the map ${\\rm Res}_D\\colon \\Omega^\\bullet(\\mathcal{A}) \\to \\Omega^{\\bullet-\\ell}(\\mathcal{B}; \\wedge^\\ell E^*)$, where $\\ell = \\dim \\ker \\rho_\\mathcal{A}|_D = {\\rm rank}(\\mathcal{A}) - {\\rm rank}(\\mathcal{B})$.\n\\end{defn}\nThere are also lower residues ${\\rm Res}_{-m}\\colon \\wedge^k W^* \\to \\mathcal{F}^{\\ell+1} \/ \\mathcal{F}^{\\ell-m}$ for $m > 0$. These are always defined but have a better description for forms $\\rho \\in \\wedge^k W^*$ whose higher residues vanish, so that ${\\rm Res}_{-m}(\\rho) \\in \\mathcal{F}^{\\ell-m +1} \/ \\mathcal{F}^{\\ell -m} \\cong \\wedge^{k-\\ell+m} V^* \\otimes \\wedge^{\\ell-m} E^*$. This leads to the following.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $D \\subseteq X$ a projective $\\mathcal{A}$-invariant submanifold with image $\\mathcal{B}$ and germinal isotropy $E := \\ker \\widetilde{\\rho}_\\mathcal{A}|_D$. The \\emph{lower residue maps} for $m \\geq 0$ of $\\mathcal{A}$ along $D$ are ${\\rm Res}_{D,-m}\\colon \\Omega^\\bullet_{1-m}(\\mathcal{A}) \\to \\Omega^{\\bullet - \\ell + m}(\\mathcal{B}; \\wedge^{\\ell - m} E^*)$, where $\\Omega^\\bullet_{1-m}(\\mathcal{A})$ is inductively the space of $\\mathcal{A}$-forms all of whose $(1-m)$th or higher residues along $D$ vanish.\n\\end{defn}\n\\begin{rem} In case $D = Z_\\mathcal{A}$ for a Lie algebroid $\\mathcal{A}$ with dense isomorphism locus, one should think of these residues as extracting the coefficients in front of the singular parts of an $\\mathcal{A}$-form along $Z_\\mathcal{A}$, as then $(\\ker \\widetilde{\\rho}_\\mathcal{A})^*|_D$ consists of ``singular'' generators.\n\\end{rem}\n\\begin{rem} Residue maps arise out of any extension of Lie algebroids. This extension need not be induced by the anchor map of the Lie algebroid, even though this is typical.\n\\end{rem}\nGiven a map of short exact sequences $\\Psi\\colon \\widetilde{\\mathcal{S}} \\to \\mathcal{S}$ with dual map $\\Psi^*\\colon \\mathcal{S}^* \\to \\widetilde{\\mathcal{S}}^*$, there is a corresponding map of filtrations $\\Psi^*\\colon \\mathcal{F}^i \\to \\widetilde{\\mathcal{F}}^i$. Setting $\\widetilde{\\ell} := \\dim \\widetilde{E}$, we have the following.\n\\begin{lem}\\label{lem:residues} In the above setting, assume that $\\widetilde{\\ell} > \\ell$. Then $\\widetilde{\\rm Res}(\\Psi^* \\rho) = 0$ for all $\\rho \\in \\wedge^k W^*$.\n\\end{lem}\n\\bp We have $\\rho \\in \\mathcal{F}^{\\ell +1}$ so that $\\Psi^* \\rho \\in \\widetilde{\\mathcal{F}}^{\\ell + 1}$. As $\\widetilde{\\ell} > \\ell$, we have $\\widetilde{\\mathcal{F}}^{\\ell+1} \\subset \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}} \\subset \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}+1}$, so that $\\widetilde{\\rm Res}(\\Psi^* \\rho) = [\\Psi^* \\rho] \\in \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}+1} \/ \\widetilde{\\mathcal{F}}^{\\widetilde{\\ell}}$ vanishes by degree reasons as desired.\n\\end{proof}\nAssuming $\\widetilde{\\ell} > \\ell$, we see from \\autoref{lem:residues} that all lower residues automatically vanish by degree reasons until considering $\\Psi^* \\rho \\in \\widetilde{\\mathcal{F}}^{\\ell +1}$. Hence the first possibly nonzero residue is $\\widetilde{\\rm Res}_{\\ell-\\widetilde{\\ell}}(\\Psi^* \\rho) = [\\Psi^* \\rho] \\in \\widetilde{\\mathcal{F}}^{\\ell+1} \/ \\widetilde{\\mathcal{F}}^{\\ell} \\cong \\wedge^{k-\\ell} \\widetilde{V}^* \\otimes \\wedge^\\ell \\widetilde{E}^*$. As in \\autoref{lem:residues} we obtain the following.\n\\begin{lem}\\label{lem:residuecommute} In the above setting, assuming $\\widetilde{\\ell} \\geq \\ell$, we have $\\Psi^* \\circ {\\rm Res} = \\widetilde{\\rm Res}_{\\ell - \\widetilde{\\ell}} \\circ \\Psi^*$.\n\\end{lem}\nOut of the above discussion the following result on residue maps is immediate.\n\\begin{prop}\\label{prop:ainvsubmfdresidue} Let $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ be a Lie algebroid morphism and $D \\subseteq X^n$, $D' \\subseteq X'$ transitive $\\mathcal{A}$- respectively $\\mathcal{A}'$-invariant submanifolds such that $f\\colon (X,D) \\to (X',D')$ is a strong map of pairs. Define $\\ell = {\\rm rank}(\\mathcal{A}) - \\dim D$ and $\\ell' = {\\rm rank}(\\mathcal{A}') - \\dim D'$, and assume that $\\ell' \\geq \\ell$. Then ${\\rm Res}_{\\ell - \\ell'} \\circ \\varphi^* = f^* \\circ {\\rm Res}$. Moreover, ${\\rm Res}_{-m} \\circ \\varphi^* = 0$ for $m > \\ell' - \\ell$.\n\\end{prop}\n\\bp As $f$ is a strong map of pairs, we have $T f\\colon TD \\to TD'$. Moreover, as noted below \\autoref{defn:laisotropy}, $\\varphi$ restricts to $\\varphi\\colon \\ker \\rho_\\mathcal{A}|_D \\to \\ker \\rho_{\\mathcal{A}'}|_{D'}$. Consequently, $\\varphi$ induces a map of the relevant short exact sequences defining the residues. The result follows from \\autoref{lem:residuecommute}.\n\\end{proof}\nFor Lie algebroids of divisor-type this specifies to the following.\n\\begin{cor}\\label{cor:residuemapsdenseiso} Let $(\\varphi,f)\\colon (\\mathcal{A},X) \\to (\\mathcal{A}',X')$ be a Lie algebroid morphism between Lie algebroids of divisor-type with transitive degeneracy loci. Assume that $f\\colon (X,Z_\\mathcal{A}) \\to (X',Z_{\\mathcal{A}'})$ is a strong map of pairs and that ${\\rm codim}\\, D' \\geq {\\rm codim}\\, D$. Then ${\\rm Res}_{\\ell-\\ell'} \\circ \\varphi^* = f^* \\circ {\\rm Res}$ and ${\\rm Res}_{-m} \\circ \\varphi^* = 0$ for $m > \\ell' - \\ell$, where $\\ell = {\\rm codim}\\, D$ and $\\ell' = {\\rm codim}\\, D'$.\n\\end{cor}\n\\bp This follows from \\autoref{prop:ainvsubmfdresidue}, where we use that degeneracy loci are $\\mathcal{A}$-invariant due to \\autoref{prop:degenlocus}, and that ${\\rm rank}(\\mathcal{A}) = \\dim X$ because $X_\\mathcal{A}$ is nonempty.\n\\end{proof}\n\\begin{rem} When we speak of projective or transitive degeneracy loci, it is implied that we assume that each degeneracy locus has components being smooth submanifolds.\n\\end{rem}\n\\subsection{Lie algebroid modules and cohomology}\n\\label{sec:aconnareps}\nWe next discuss Lie algebroid connections and Lie algebroid modules, which are vector bundles equipped with a flat Lie algebroid connection. We further introduce the associated Lie algebroid cohomologies and characteristic classes, including the modular class. For more information, see e.g.\\ \\cite[Chapter 7]{Mackenzie05} and also \\cite{EvensLuWeinstein99, KosmannSchwarzbachLaurentGengouxWeinstein08,GualtieriPym13}. Let $\\mathcal{A} \\to X$ be a Lie algebroid and $E \\to X$ a vector bundle.\n\\begin{defn} An \\emph{$\\mathcal{A}$-connection} on $E$ is a bilinear map $\\nabla\\colon \\Gamma(\\mathcal{A}) \\times \\Gamma(E) \\to \\Gamma(E)$, $(v,\\sigma) \\mapsto \\nabla_v \\sigma$, such that $\\nabla_{f v} \\sigma = f \\nabla_v \\sigma$ and $\\nabla_v(f\\sigma) = f \\nabla_v \\sigma + (\\rho_\\mathcal{A}(v) f) \\cdot \\sigma$ for all $v \\in \\Gamma(\\mathcal{A})$, $\\sigma \\in \\Gamma(E)$ and $f \\in C^\\infty(X)$. We will also consider $\\nabla$ as a map $\\nabla\\colon \\Gamma(E) \\to \\Omega^1(\\mathcal{A};E)$ via $(\\nabla s)(v) := \\nabla_v s$.\n\\end{defn}\nAny $\\mathcal{A}$-connection $\\nabla$ on $E$ has a \\emph{curvature tensor} $F_\\nabla \\in \\Gamma(\\wedge^2 \\mathcal{A}^* \\otimes {\\rm End}(E))$, given by $F_\\nabla(v,w) := \\nabla_v \\circ \\nabla_w - \\nabla_w \\circ \\nabla_v - \\nabla_{[v,w]_\\mathcal{A}}$ for $v, w \\in \\Gamma(\\mathcal{A})$. An $\\mathcal{A}$-connection $\\nabla$ is \\emph{flat} if $F_\\nabla \\equiv 0$.\n\\begin{defn}\\label{defn:amodule} An \\emph{$\\mathcal{A}$-module} is a vector bundle $E$ equipped with a flat $\\mathcal{A}$-connection $\\nabla$.\n\\end{defn}\nGiven an $\\mathcal{A}$-module $(E,\\nabla)$, there is a differential $d_{\\mathcal{A},\\nabla}$ on $\\Omega^\\bullet(\\mathcal{A};E)$, the space of $\\mathcal{A}$-forms with values in $E$. Recalling that $\\Omega^\\bullet(\\mathcal{A};E) = \\Gamma(\\wedge^\\bullet \\mathcal{A}^*) \\otimes \\Gamma(E)$, we set $d_{\\mathcal{A},\\nabla}(\\eta \\otimes s) := d_{\\mathcal{A}} \\eta \\otimes s + (-1)^{|\\eta|} \\eta \\otimes \\nabla s$, with $|\\eta|$ the degree of $\\eta$. This differential satisfies $d_{\\mathcal{A},\\nabla}(\\eta \\wedge \\xi \\otimes s) = d_{\\mathcal{A}} \\eta \\wedge \\xi \\otimes s + (-1)^{|\\eta|} \\eta \\wedge dt_{\\mathcal{A},\\nabla} (\\xi \\otimes s)$, and $d_{\\mathcal{A},\\nabla}$ squares to zero if and only if $\\nabla$ is flat. In fact, there is a bijective correspondence between such operators on $\\Omega^\\bullet(\\mathcal{A};E)$, and flat $\\mathcal{A}$-connections on $E$.\n\\begin{defn} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $(E,\\nabla)$ an $\\mathcal{A}$-module. The \\emph{$\\mathcal{A}$-cohomology with values in $E$} is given by $H^k(\\mathcal{A}; E) := H^k(\\Omega^\\bullet(\\mathcal{A};E), d_{\\mathcal{A},\\nabla})$ for $k \\in \\mathbb{N} \\cup \\{0\\}$.\n\\end{defn}\nNote that $\\mathcal{A}$-modules are also called \\emph{$\\mathcal{A}$-representations}, and that they are Lie algebra representations when $X = \\{{\\rm pt}\\}$. Denote the space of all $\\mathcal{A}$-representations by $\\mathcal{A}$-${\\rm Rep}(X)$.\n\\begin{exa} Let $\\underline{\\mathbb{R}} \\to X$ be the trivial line bundle, carrying a trivial $\\mathcal{A}$-representation structure given by $\\nabla_v f := \\mathcal{L}_{\\rho_{\\mathcal{A}}(v)} f$ for $v \\in \\Gamma(\\mathcal{A})$ and $f \\in \\Gamma(\\underline{\\mathbb{R}}) = C^\\infty(X)$. We then have that $H^\\bullet(\\mathcal{A};\\underline{\\mathbb{R}}) =: H^\\bullet(\\mathcal{A})$, obtaining what is usually called the \\emph{Lie algebroid cohomology} of $\\mathcal{A}$.\n\\end{exa}\nLie algebroid cohomology is generally hard to compute and need not be finite-dimensional. However, the cohomology of most of the Lie algebroids of divisor-type can be determined (see \\cite{KlaasseLanius18} for an overview). Any Lie algebroid $\\mathcal{A} \\to X$ always has a canonical $\\mathcal{A}$-module (see \\cite{EvensLuWeinstein99}).\n\\begin{defn}\\label{defn:canamodule} Let $\\mathcal{A} \\to X$ be a Lie algebroid and set $Q_\\mathcal{A} := \\det(\\mathcal{A}) \\otimes \\det(T^*X)$. Then $Q_\\mathcal{A}$ is  the \\emph{canonical $\\mathcal{A}$-module}, using the $\\mathcal{A}$-connection on $Q_\\mathcal{A}$ that is defined for $v \\in \\Gamma(\\mathcal{A})$, $V \\in \\Gamma(\\det(\\mathcal{A}))$ and $\\mu \\in \\Gamma(\\det(T^*X))$ by the formula $\\nabla_v(V \\otimes \\mu) := \\mathcal{L}_{v} V \\otimes \\mu + V \\otimes \\mathcal{L}_{\\rho_\\mathcal{A}(v)} \\mu$.\n\\end{defn}\nOne readily checks this formula indeed defines a flat $\\mathcal{A}$-connection on the line bundle $Q_\\mathcal{A}$. Several operations can be performed on the class of Lie algebroid representations, such as pullbacks along Lie algebroid morphisms, duals, tensor products, and exterior products.\n\\begin{rem} For Lie algebroids $\\mathcal{A}^n \\to X$ of divisor-type we have $Q_\\mathcal{A}^* \\cong {\\rm div}(\\mathcal{A})$. There should be a relation between this and the Evens--Lu--Weinstein pairing $H^k(\\mathcal{A}) \\otimes H^{n-k}(\\mathcal{A};Q_\\mathcal{A}) \\to \\mathbb{R}$ (see \\cite{EvensLuWeinstein99}), especially when $\\mathcal{A} = TX_I$ for a standard projective divisor ideal $I \\subseteq C^\\infty(X)$.\n\\end{rem}\nAny $\\mathcal{A}$-module $(L,\\nabla)$ of rank $1$ induces a cohomology class in $H^1(\\mathcal{A})$. Given a nonvanishing section $s \\in \\Gamma(L^2)$ and $v \\in \\Gamma(\\mathcal{A})$ (noting that $L^2 = L \\otimes L$ is always trivial) we can write $\\langle v, d_{\\mathcal{A},\\nabla} s\\rangle = \\theta_s(v) s$ for some element $\\theta_s \\in \\Gamma(\\mathcal{A}^*)$. As $d_{\\mathcal{A},\\nabla}$ squares to zero, the section $\\theta_s$ is $d_\\mathcal{A}$-closed, and one verifies that the cohomology class $\\theta_L := \\frac12 [\\theta_s] \\in H^1(\\mathcal{A})$ is independent of $s$. This is the \\emph{characteristic class} of $(L,\\nabla)$. Characteristic classes are natural with respect to tensor products and pullbacks of modules, i.e.\\ they satisfy $\\theta_{L \\otimes L'} = \\theta_L + \\theta_{L'}$ and $\\varphi^* \\theta_L = \\theta_{\\varphi^*L} \\in H^1(\\mathcal{A}')$ given $\\mathcal{A}$-modules $L$ and $L'$ of rank $1$ and a Lie algebroid morphism $\\varphi\\colon \\mathcal{A}' \\to \\mathcal{A}$.\n\nThe characteristic class of the trivial $\\mathcal{A}$-module is zero, and the characteristic class of $Q_\\mathcal{A}$ is called the \\emph{modular class} ${\\rm mod}(\\mathcal{A})$ of $\\mathcal{A}$. One says that $\\mathcal{A}$ is \\emph{unimodular} if ${\\rm mod}(\\mathcal{A}) = 0$. Similarly an $\\mathcal{A}$-module $(L,\\nabla)$ of rank $1$ is \\emph{umimodular} if its characteristic class vanishes. In this case it admits a global nonvanishing section which is $d_{\\mathcal{A},\\nabla}$-closed.\n\\subsubsection{Representations up to homotopy}\nIn studying the residue map of \\autoref{sec:residuemaps} we will have use for a powerful extension of the notion of an $\\mathcal{A}$-representation, introduced in \\cite{AriasAbadCrainic12}. To start, recall that a \\emph{$\\mathbb{Z}$-graded vector bundle} $\\mathbb{E} \\to X$ is a direct sum $\\mathbb{E} = \\bigoplus_{i \\in \\mathbb{Z}} E_i$ of vector bundles, where $E_i$ has \\emph{degree} $i$. Further, ${\\rm End}_k(\\mathbb{E}) \\to X$ is the associated bundle of endomorphisms of $\\mathbb{E}$ of degree $k$. Let $\\mathcal{A} \\to X$ be a Lie algebroid. Then $\\Omega^\\bullet(\\mathcal{A};\\mathbb{E}) := \\Gamma(\\wedge^\\bullet \\mathcal{A}^* \\otimes \\mathbb{E})$ becomes a right graded $\\Omega^\\bullet(\\mathcal{A})$-module in the standard way. With this we have:\n\\begin{defn}[\\cite{AriasAbadCrainic12}] A $\\mathbb{Z}$-graded vector bundle $\\mathbb{E} \\to X$ is an \\emph{$\\mathcal{A}$-representation up to homotopy} if $\\Omega^\\bullet(\\mathcal{A};\\mathbb{E})$ carries a differential $D$ such that it is a right graded $(\\Omega^\\bullet(\\mathcal{A}),d_\\mathcal{A})$-module.\n\\end{defn}\nAs for $\\mathcal{A}$-representations, these lead to cohomology theories $H^\\bullet(\\mathcal{A};\\mathbb{E})$. Moreover, there is a similar bijective correspondence between such differentials $D$, and $\\mathcal{A}$-connections on $\\mathbb{E}$ coming with maps $\\omega_i\\colon \\Gamma(\\wedge^i \\mathcal{A}) \\to {\\rm End}_{1-i}(\\mathbb{E})$ for $i \\geq 2$ satisfying certain coherence conditions.\n\nDenote by $\\mathcal{A}$-${\\rm RepHtpy}$ the space of $\\mathcal{A}$-representations up to homotopy, with $\\mathcal{A}$-${\\rm RepHtpy}_{k,k'}$ for $k,k' \\in \\mathbb{Z}$ the subclasses of those for which the associated $\\mathbb{Z}$-graded vector bundle $\\mathbb{E}$ satisfies $E_i = \\{0\\}$ for all $i$ not satisfying $k \\leq i < k'$. Finally, set $\\mathcal{A}$-${\\rm RepHtpy}_{k,k} := \\mathcal{A}$-${\\rm RepHtpy}_{k,k}$ for those concentrated in degree $k$. Note then that there is a natural inclusion $\\mathcal{A}$-${\\rm Rep} \\hookrightarrow \\mathcal{A}$-${\\rm RepHtpy}_0$ of $E \\mapsto \\mathbb{E}$ with $\\mathbb{E}_0 = E$. The class of $\\mathcal{A}$-representations up to homotopy admits duals, tensor products, and exterior products, similar to what is true for $\\mathcal{A}$-representations.\n\nWe now explain that $\\mathcal{A}$-representations up to homotopy give rise to $\\mathcal{A}$-representations. Given a $\\mathbb{Z}$-graded vector bundle $\\mathbb{E} \\to X$, let $\\mathbb{E}_{\\rm red}$ be the resulting $\\mathbb{Z}_2$-graded vector bundle obtained from the map $\\mathbb{Z} \\to \\mathbb{Z}_2$ of reduction modulo two. Given such a $\\mathbb{Z}_2$-graded vector bundle $\\mathbb{F} \\to X$ with summands $\\mathbb{F} = F_0 \\oplus F_1$, we can consider its \\emph{Berizinian}\n\\begin{equation*}\n\t{\\rm Ber}(F) := \\det(F_0) \\otimes \\det(F_1^*) \\to X.\n\\end{equation*}\nWith this, given a $\\mathbb{Z}$-graded vector bundle $\\mathbb{E} \\to X$, define ${\\rm Ber}(\\mathbb{E}) := {\\rm Ber}(\\mathbb{E}_{\\rm red}) \\to X$.\n\\begin{prop}[\\cite{Mehta15}]\\label{prop:berrephtpy} Let $\\mathbb{E} \\in \\mathcal{A}$-${\\rm RepHtpy}$. Then assigning $\\mathbb{E} \\mapsto {\\rm Ber}(\\mathbb{E})$ defines a map\n\\begin{equation*}\n\t{\\rm Ber}\\colon \\mathcal{A}\\text{-}{\\rm RepHtpy} \\to \\mathcal{A}\\text{-}{\\rm Rep}.\n\\end{equation*}\n\\end{prop}\n\\begin{rem}\\label{rem:adjointrep} We noted in \\autoref{defn:canamodule} that there is a canonical modular $\\mathcal{A}$-module $Q_\\mathcal{A} = \\det(\\mathcal{A}) \\otimes \\det(T^*X)$. We can understand this better using the above framework. Namely, every Lie algebroid $\\mathcal{A} \\to X$ has a canonical $\\mathcal{A}$-representation up to homotopy, namely its \\emph{adjoint representation (up to homotopy)} ${\\rm Adj}(\\mathcal{A})$, given by the anchor sequence $\\mathcal{A} \\to TX$ with its natural differential $D_\\nabla$ \\cite{AriasAbadCrainic12}. Note now that ${\\rm Ber}({\\rm Adj}(\\mathcal{A})) = Q_\\mathcal{A}$, and one readily checks that the resulting induced $\\mathcal{A}$-module structure on $Q_\\mathcal{A}$ is the one of \\autoref{defn:canamodule} (see \\cite{Mehta15}).\n\\end{rem}\n\\begin{rem} The cohomology groups $H^\\bullet(\\mathcal{A}; {\\rm Adj}(\\mathcal{A}))$ of the adjoint representation of \\autoref{rem:adjointrep} form the \\emph{deformation cohomology} $H^\\bullet_{\\rm def}(\\mathcal{A})$ of the Lie algebroid $\\mathcal{A} \\to X$ \\cite{CrainicMoerdijk08}. This important fact motivated the very definition of an $\\mathcal{A}$-representation up to homotopy (see \\cite{AriasAbadCrainic12}).\n\\end{rem}\nThere is a class of representations up to homotopy called \\emph{Serre representations} that is of main importance to us (\\cite[Example 4.15]{AriasAbadCrainic12}). Consider an extension of Lie algebroids over a manifold $X$, i.e.\\ a short exact sequence of Lie algebroid morphisms as follows:\n\\begin{equation*}\n\t0 \\to E \\stackrel{i}{\\to} \\mathcal{A} \\stackrel{p}{\\to} \\mathcal{A}' \\to 0.\n\\end{equation*}\nLet $\\sigma\\colon \\Gamma(\\mathcal{A}') \\to \\Gamma(\\mathcal{A})$ be an Ehresmann connection splitting the above sequence, i.e.\\ such that $p \\circ \\sigma = {\\rm id}_{\\mathcal{A}'}$. This gives a $\\sigma$-dependent identification $\\Gamma(\\mathcal{A}) \\cong \\Gamma(\\mathcal{A}') \\oplus \\Gamma(E)$. Consider the $\\mathbb{Z}$-graded complex $(\\Gamma(\\wedge^\\bullet E^*), d_E)$. Given $v \\in \\Gamma(\\mathcal{A}')$ there is a degree-zero operator $\\nabla_v^\\sigma := {\\rm ad}^*_{\\sigma(v)}\\colon \\Gamma(\\wedge^\\bullet E^*) \\to \\Gamma(\\wedge^\\bullet E^*)$ using the bracket $[\\cdot,\\cdot]_\\mathcal{A}$, which is thus an $\\mathcal{A}'$-connection $\\nabla^\\sigma$ on $\\Gamma(\\wedge^\\bullet E^*)$. Further, the \\emph{curvature} $R_\\sigma \\in \\Gamma(\\wedge^2 \\mathcal{A}'^* \\otimes E)$ of $\\sigma$ is given for $v,v' \\in \\Gamma(\\mathcal{A}')$ by\n\\begin{equation*}\n\tR^\\sigma(v,v') := [\\sigma(v),\\sigma(v')]_\\mathcal{A} - \\sigma([v,v']_{\\mathcal{A}'}).\n\\end{equation*}\nThis naturally turns into a map $i(R^\\sigma)\\colon \\Gamma(\\wedge^2 \\mathcal{A}') \\to {\\rm End}_{-1}(\\wedge^\\bullet E^*)$ after using the contraction $i\\colon E \\to {\\rm End}_{-1}(\\wedge^\\bullet E^*)$. Now define the operator $D := d_E + \\nabla^\\sigma + i(R^\\sigma)$ on $\\mathbb{E} := \\wedge^\\bullet E^*$.\n\\begin{prop}[{\\cite[Example 4.15]{AriasAbadCrainic12}}]\\label{prop:rephtpyextension} The triple $(\\mathbb{E},\\sigma, D)$ defines the structure of an $\\mathcal{A}'$-representation up to homotopy on the exterior algebra $\\mathbb{E} = \\wedge^\\bullet E^*$.\n\\end{prop}\n\\begin{rem} When $E$ is abelian, i.e.\\ has trivial bracket, the above structure in fact defines an $\\mathcal{A}$-module structure on $E$. More generally this is true for the center and abelianization $Z(E)$ and $E\/[E,E]$ of the bundle Lie algebroid $E$ (see \\cite[Proposition 3.3.20]{Mackenzie05}).\n\\end{rem}\n\\subsubsection{Residue maps and representations}\nWe now discuss when the residue maps of \\autoref{sec:residuemaps} induce maps in cohomology. Let $\\mathcal{A} \\to X$ be a Lie algebroid and let $D \\subseteq X$ a projective $\\mathcal{A}$-invariant submanifold, so that we obtain an extension of Lie algebroids\n\\begin{equation*}\n\t0 \\to \\ker\\widetilde{\\rho}_{\\mathcal{A}}|_D \\to \\mathcal{A}|_D \\to \\mathcal{B} \\to 0.\n\\end{equation*}\nBy applying \\autoref{prop:rephtpyextension} we obtain the structure of a $\\mathcal{B}$-representation up to homotopy on $E := \\ker \\widetilde{\\rho}_\\mathcal{A}|_D \\to D$. Using \\autoref{prop:berrephtpy} we obtain a $\\mathcal{B}$-module structure on $\\det(E) \\to D$, and consequently also on its dual $\\det(E^*)$. We can now prove the following (\\autoref{thm:introresidues}).\n\\begin{thm}\\label{thm:residues} Let $\\mathcal{A} \\to X$ be a Lie algebroid and $(\\mathcal{B},D)$ a projective $\\mathcal{A}$-invariant submanifold with $\\ell$-dimensional germinal isotropy $E \\to D$. Then ${\\rm Res}_D\\colon \\Omega^\\bullet(X,\\mathcal{A}) \\to \\Omega^{\\bullet-\\ell}(D, \\mathcal{B}; \\det(E^*))$ is a cochain morphism, hence induces a map $[{\\rm Res}_D]\\colon H^\\bullet(X,\\mathcal{A}) \\to H^{\\bullet-\\ell}(D,\\mathcal{B};\\det(E^*))$.\n\\end{thm}\n\\bp The $\\mathcal{B}$-representation on $\\det(E)$ is induced from the $\\mathcal{B}$-connection $\\nabla^\\sigma = [\\sigma(\\cdot),\\cdot]_\\mathcal{A}$. The condition that ${\\rm Res}_D$ is a cochain morphism is equivalent to the fact that $\\det(E^*) \\subseteq \\wedge^\\bullet \\mathcal{A}^*|_D$ has closed local generating sections. This is implied by \\autoref{prop:berrephtpy}, giving the result.\n\\end{proof}\n\\begin{rem} The above theorem is not true in general for the lower residue maps discussed in \\autoref{sec:residuemaps}. Counterexamples arise from the local generators of $\\wedge^{\\ell-k} E^*$ not being closed. However, by analyzing the proof of \\autoref{thm:residues} we see that it is true for ${\\rm Res}_{D,-k}$ whenever the representation up to homotopy $\\wedge^{\\ell-k} E$ is a $\\mathcal{B}$-module. This property is inherited (i.e.\\ if it holds for some $k$, then also for all $k' \\leq k$), and the above shows that this is true for $k = 0$.\n\\end{rem}\nTo better understand the Lie algebroid cohomology acting as the target for the residue map, it is clear that it is important to determine when, or whether, the $\\mathcal{B}$-representation on $\\det(E)$ is unimodular, so that one can suitably trivialize $\\det(E)$ as a representation.\n\\begin{rem}\\label{rem:residuesymplecticliealgebroid} Let $(\\mathcal{A},\\omega_{\\mathcal{A}}) \\to X$ be a symplectic Lie algebroid and let $D \\subseteq X$ be a transitive $\\mathcal{A}$-invariant submanifold, possibly the degeneracy locus of $\\mathcal{A}$ (see \\autoref{prop:degenlocus}). The residue map gives a form ${\\rm Res}_D(\\omega_\\mathcal{A}) \\in \\Omega^{2-\\ell}(D; \\wedge^\\ell E^*)$, where $E = \\ker \\rho_{\\mathcal{A}}|_D$ and $\\ell = \\dim(E)$. Note however that it can only be nonzero for $\\ell \\leq 2$. One further has the residue ${\\rm Res}_D(\\omega_{\\mathcal{A}}^n) \\in \\Omega^{2n-\\ell}(D;\\wedge^\\ell E^*)$, if ${\\rm rank}(\\mathcal{A}) = 2n$. These describe important features of the induced geometry on $D$, and the latter has a Poisson-geometric interpretation which we discuss in \\autoref{sec:residuesympla}.\n\\end{rem}\n\\begin{rem} While \\autoref{thm:residues} shows the residue map descends to cohomology, it often does not respect the ring structure induced by the wedge product. See also \\autoref{sec:residuesympla}.\n\\end{rem}\n\\subsubsection{Examples of residue maps}\n\\label{sec:residueexamples}\nIn this section we discuss several examples of residue maps, for some of the Lie algebroids of divisor-type that we have discussed in this paper.\n\n\\begin{exa}[$\\mathcal{A}_Z^k$]\\label{exa:bkersiduemaps} For the $b^k$-bundles $\\mathcal{A}_Z^k \\to X$ of \\autoref{exa:bktangentbundle}, the hypersurface $Z \\subseteq X$ is a transitive invariant submanifold. The resulting extension sequence of Lie algebroids\n\\begin{equation*}\n\t0 \\to \\mathbb{L}_{Z,k} \\to \\mathcal{A}_Z^k|_Z \\to TZ \\to 0,\n\\end{equation*}\nis such that $\\mathbb{L}_{Z,k} \\to Z$ is trivial, and has a canonical trivializing section (it is also called the \\emph{$b^k$-normal bundle}, see \\cite[Proposition 4]{GuilleminMirandaPires14} and \\cite[Proposition 3.2]{Scott16}). As such the residue map is a cochain morphism, given by ${\\rm Res}_{Z,k}\\colon \\Omega^\\bullet(\\mathcal{A}_Z^k) \\to \\Omega^{\\bullet-1}(Z)$, and locally $\\frac{dx}{x^k} \\wedge \\alpha + \\beta \\mapsto \\iota_Z^*(\\alpha)$, where $x \\in j_{k-1}$ is a local $(k-1)$-jet generator, $\\alpha,\\beta \\in \\Omega^\\bullet(X)$ and $\\iota_Z\\colon Z \\hookrightarrow X$ is the inclusion. These residue maps along with the dual $\\mathcal{A}_Z^{k-1}$-anchor $\\varphi_{A_Z^k}^*$ fit in the short exact sequence\n\\begin{equation*}\n\t0 \\to \\Omega^\\bullet(\\mathcal{A}_Z^{k-1}) \\to \\Omega^\\bullet(\\mathcal{A}_Z) \\to \\Omega^{\\bullet-1}(Z) \\to 0,\n\\end{equation*}\nwhich is called the \\emph{residue sequence} for $\\mathcal{A}_Z^k$. These sequences split and imply the cohomological consequences $H^\\bullet(\\mathcal{A}_Z^k) \\cong H^\\bullet(\\mathcal{A}_Z^{k-1}) \\oplus H^{\\bullet-1}(Z)$ (c.f.\\ \\cite{GuilleminMirandaPires14,MarcutOsornoTorres14,Melrose93,Scott16}; see \\cite{KlaasseLanius18} for more).\n\\end{exa}\n There are no lower residue maps for $\\mathcal{A}_Z^k \\to X$, as $\\mathbb{L}_{Z,k}$ is one-dimensional. While \\autoref{thm:residues} asserts the existence of the above residue map as stated, it is special in this situation that $\\mathbb{L}_{Z,k}$ is not only (canonically) trivial, but is trivial as a $TZ$-representation, so that the image of the residue map is smooth forms on $Z$, i.e.\\ $\\Omega^{\\bullet-1}(Z)$, instead of $\\Omega^{\\bullet-1}(Z;\\mathbb{L}_{Z,k}^*)$.\n\\begin{exa}[$\\mathcal{A}_{|D|}$]\\label{exa:elltangentresidues} For the elliptic tangent bundle $\\mathcal{A}_{|D|} \\to X$ of \\autoref{exa:ellvx}, the degeneracy locus $D \\subseteq X$ is a transitive invariant submanifold, hence leads to an extension (\\cite{CavalcantiGualtieri18})\n\\begin{equation*}\n\t0 \\to \\underline{\\mathbb{R}} \\oplus \\mathfrak{k} \\to \\mathcal{A}_{|D|}|_D \\to TD \\to 0.\n\\end{equation*}\nNote that $\\ker(\\rho_{\\mathcal{A}_{|D|}}) \\cong \\underline{\\mathbb{R}} \\oplus \\mathfrak{k}$, so that $\\det(\\ker(\\rho_{\\mathcal{A}_{|D|}})) \\cong \\mathfrak{k}$. This results in an \\emph{elliptic residue}\n\\begin{equation*}\t\n\t{\\rm Res}_q\\colon \\Omega^\\bullet(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-2}(D;\\mathfrak{k}^*), \\qquad d\\log r \\wedge d\\theta \\wedge \\alpha + d\\log r \\wedge \\beta + d\\theta \\wedge \\gamma + \\eta \\mapsto \\iota_D^*(\\alpha),\n\\end{equation*}\nfor $\\alpha,\\beta,\\gamma,\\eta \\in \\Omega^\\bullet(X)$ and $\\iota_D\\colon D \\hookrightarrow X$ the inclusion. Letting $\\Omega^\\bullet_0(\\mathcal{A}_{|D|}) = \\ker({\\rm Res}_q)$ we have\n\\begin{equation*}\n\t{\\rm Res}_r\\colon \\Omega^\\bullet_0(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-1}(D), \\qquad d\\log r \\wedge \\beta + d\\theta \\wedge \\gamma + \\eta \\mapsto \\iota_D^*(\\beta),\n\\end{equation*}\nwhich is called the \\emph{radial residue}. A coorientation for $D$ trivializes $\\mathfrak{k}$ and gives further\n\\begin{equation*}\n\t{\\rm Res}_\\theta\\colon \\Omega^\\bullet_0(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-1}(D), \\qquad d\\log r \\wedge \\beta + d\\theta \\wedge \\gamma + \\eta \\mapsto \\iota_D^*(\\gamma),\n\\end{equation*}\nwhich is the \\emph{$\\theta$-residue}. In general ${\\rm Res}_r\\colon \\Omega^\\bullet(\\mathcal{A}_{|D|}) \\to \\Omega^{\\bullet-1}(D,{\\rm At}(S^1 ND))$ and ${\\rm Res}_q = \\iota_{\\partial_{\\theta}} \\circ {\\rm Res}_r$. This is because $\\mathcal{A}_{|D|}|_D$ is an extension by $\\underline{\\mathbb{R}}$ of the Atiyah algebroid ${\\rm At}(S^1 ND) \\to D$, i.e.\\\n\\begin{equation*}\n\t0 \\to \\underline{\\mathbb{R}} \\to \\mathcal{A}_{|D|}|_D \\to {\\rm At}(S^1 ND) \\to 0,\n\\end{equation*}\nso that by the general recipe for residue maps we obtain the map ${\\rm Res}_r$ above, as ${\\rm det}(\\underline{\\mathbb{R}}^*)$ is the trivial representation. Finally ${\\rm At}(S^1 ND)$ has $D$ as transitive invariant submanifold, with\n\\begin{equation*}\n\t0 \\to \\mathfrak{k} \\to {\\rm At}(S^1 ND) \\to TD \\to 0,\n\\end{equation*}\nwhose residue map ${\\rm Res}_{\\rm At}\\colon \\Omega^\\bullet(D, {\\rm At}(S^1 ND)) \\to \\Omega^{\\bullet-1}(D; \\mathfrak{k}^*)$ combines as ${\\rm Res}_q = {\\rm Res}_{\\rm At} \\circ {\\rm Res}_r$, and ${\\rm Res}_{\\rm At} = \\iota_{\\partial_{\\theta}}$ in local coordinates.\nAs in \\autoref{exa:bkersiduemaps} there are cohomological consequences one can draw from this (see \\cite{CavalcantiGualtieri18,CavalcantiKlaasse18}), for example $H^\\bullet(\\mathcal{A}_{|D|}) \\cong H^\\bullet(X\\backslash D) \\oplus H^{\\bullet-1}(S^1 ND)$.\n\\end{exa}\nThe previous example is special as the isotropy bundle is two-dimensional, but splits as a sum of two line bundles. As such we have ${\\rm Res}_{D,0} = {\\rm Res}_q$ and ${\\rm Res}_{D,-1} =  {\\rm Res_r} + {\\rm Res}_\\theta$ here. There are also residue maps for complex Lie algebroids, for example the complex log-tangent bundle $\\mathcal{A}_D$ of \\autoref{exa:complexlogtgnt} with its complex log residue ${\\rm Res}_D\\colon \\Omega^\\bullet(\\mathcal{A}_D) \\to \\Omega^{\\bullet-1}(D;\\mathbb{C})$, which is related to the \\emph{complex residue} ${\\rm Res}_\\mathbb{C} := {\\rm Res}_{D,-1}$ of the elliptic tangent bundle $\\mathcal{A}_{|D|}$ (see \\cite{CavalcantiGualtieri18}).\n\\begin{exa}[$\\mathcal{A}_{\\underline{Z}}$]  Let $\\underline{Z} = \\cup_{j \\in I} Z_j$ be a normal-crossing log divisor with associated log-tangent bundle $\\mathcal{A}_{\\underline{Z}}$ as below \\autoref{exa:zvx}. Set $Z_\\tau = \\cap_{i \\in \\tau} Z_i$ for $\\tau \\subseteq I$ and let $\\underline{Z}_k = \\cup_{|\\tau| = k} Z_\\tau$, so that $\\underline{Z}_k$ consists of all $k$-fold intersections of hypersurfaces. By transversality, these are all smooth submanifolds, with induced normal-crossing log divisors. We have for $\\mathcal{A}_{\\underline{Z}_k} \\to \\underline{Z}_{k}$ that $\\underline{Z}_{k+1}$ is $\\mathcal{A}_{\\underline{Z}_k}$-invariant, with its associated Lie algebroid extension sequence over $\\underline{Z}_{k+1}$ being\n\\begin{equation*}\n\t0 \\to \\mathbb{L}_{\\underline{Z},k} \\to \\mathcal{A}_{\\underline{Z}_k}|_{\\underline{Z}_{k+1}} \\to \\mathcal{A}_{\\underline{Z}_{k+1}} \\to 0.\n\\end{equation*}\nAgain $\\mathbb{L}_{\\underline{Z},k} \\to \\underline{Z}_{k+1}$ is canonically trivial, leading to ${\\rm Res}_{Z,k}\\colon \\Omega^\\bullet(\\underline{Z}_k, \\mathcal{A}_{\\underline{Z}_k}) \\to \\Omega^{\\bullet-1}(\\underline{Z}_{k+1},\\mathcal{A}_{\\underline{Z}_{k+1}})$. This leads to a sequence of composable residue maps, and splitting residue sequences\n\\begin{equation*}\n\t0 \\to \\Omega^\\bullet(\\underline{Z}_k) \\to \\Omega^\\bullet(\\underline{Z}_k, \\mathcal{A}_{\\underline{Z}_k}) \\to \\Omega^{\\bullet-1}(\\mathcal{A}_{\\underline{Z}_{k+1}}) \\to 0,\n\\end{equation*}\nwhich leads to the cohomological consequence $H^\\bullet(\\underline{Z}_k,\\mathcal{A}_{\\underline{Z}_k}) \\cong H^\\bullet(\\underline{Z}_k) \\oplus H^{\\bullet-1}(\\underline{Z}_{k+1}, \\mathcal{A}_{\\underline{Z}_{k+1}})$. These are straightforward analogues of the case of the log-tangent bundle $\\mathcal{A}_Z$ of \\autoref{exa:bkersiduemaps}. By applying induction to $k$ and noting that $X = Z_0$, this gives (c.f.\\ \\cite[Appendix A.24]{GualtieriLiPelayoRatiu17})\n\\begin{equation*}\n\tH^\\bullet(\\mathcal{A}_{\\underline{Z}}) \\cong H^\\bullet(X) \\oplus \\bigoplus_{k \\geq 1} H^{\\bullet-k}(\\underline{Z}_k).\n\\end{equation*}\n\\end{exa}\nThe above can also be done in similar fashion for self-crossing log divisors (see \\cite{MirandaScott18}) instead mapping onto its $k$-strata, but this requires more notation, so that we will not do so here. The other Lie algebroids found in this paper, such as the scattering tangent bundle $\\mathcal{C}_Z \\to X$, also admit residue maps, but these are less useful, because $Z$ is $\\mathcal{C}_Z$-invariant with $\\mathcal{B} = 0_{TZ}$.\n\\begin{exa}[$\\mathcal{A}_{Z,F}$] Consider the Lie algebroid $\\mathcal{A}_{Z,F} := [TX\\:TF] \\to X$ given a codimension-one involutive distribution $TF \\subseteq TZ$ on a hypersurface $Z \\subseteq X$ (see \\autoref{sec:examplesmodification}). Then $\\mathcal{A}_{Z,F}$ has $Z$ as a projective invariant submanifold with $\\mathcal{B} = TF$ (i.e.\\ $\\mathcal{A}_{Z,F}|_Z$ is regular):\n\\begin{equation*}\n\t0 \\to \\ker(\\rho_{\\mathcal{A}_{Z,F}}|_Z) \\to \\mathcal{A}_{Z,F}|_Z \\to TF \\to 0.\n\\end{equation*}\nLocally we have $\\ker(\\rho_{\\mathcal{A}_{Z,F}}|_Z) = \\langle x \\partial_x, x \\partial_y\\rangle$ for $x \\in I_Z$ a local generator, and $\\partial_y \\subseteq \\Gamma(TZ)$ normal to $TF$. The dual generator $dx\/x$ is closed, but the generator $dy\/x$ is not. Thus, while the top residue map is a cochain morphism by \\autoref{thm:residues}, the lower residue map is not. Note that $\\mathcal{A}_{Z,F}$ is an $\\mathcal{A}_Z$-Lie algebroid, and that its $\\mathcal{A}_Z$-anchor is a quasi-isomorphism (c.f.\\ \\cite{GualtieriLiPelayoRatiu17,KlaasseLanius18}).\n\\end{exa}\nSimilar ideas apply to invariantly capture what is done in \\cite{Lanius16}, but we will not do so here.\n\\begin{exa}[$\\mathcal{A}_W$]\\label{exa:ellipticlogresidue} Let $W = Z \\otimes |D|$ be an elliptic-log divisor with associated Lie algebroid $\\mathcal{A}_W \\to X$ as in \\autoref{prop:elllogalgebroid}. The degeneracy locus $Z \\subseteq X$ is $\\mathcal{A}_W$-invariant, and $D \\subseteq Z$ is a hypersurface with log-tangent bundle $\\mathcal{A}_{Z,D} \\to Z$. There is an extension sequence\n\\begin{equation*}\n\t0 \\to E \\to \\mathcal{A}_W|_Z \\to \\mathcal{A}_{Z,D} \\to 0,\n\\end{equation*}\nwith $E$ the rank-$1$ germinal isotropy bundle. This follows from the local description of $\\Gamma(\\mathcal{A}_W) = \\langle x \\partial_x + y \\partial_y, x(y \\partial_x - x \\partial_y) \\rangle \\oplus \\Gamma(TD)$. This results in a residue map\n\\begin{equation*}\n\t{\\rm Res}_Z\\colon \\Omega^\\bullet(\\mathcal{A}_W) \\to \\Omega^{\\bullet-1}(Z,\\mathcal{A}_{Z,D};E^*).\n\\end{equation*}\nThe submanifold $D\\subseteq X$ is also $\\mathcal{A}_W$-invariant, with a transitive extension sequence\n\\begin{equation*}\n\t0 \\to \\ker(\\rho_{\\mathcal{A}_W}|_D) \\to \\mathcal{A}_W|_D \\to TD \\to 0.\n\\end{equation*}\nThis gives another residue map onto $D$, namely\n\\begin{equation*}\n\t{\\rm Res}_D\\colon \\Omega^\\bullet(\\mathcal{A}_W) \\to \\Omega^{\\bullet-2}(D;\\det(F^*)),\n\\end{equation*}\nwith isotropy plane bundle $F = \\ker(\\rho_{\\mathcal{A}_W}|_D) \\to D$. These residue maps are related, because for the log-tangent bundle $\\mathcal{A}_{Z,D} \\to Z$, the submanifold $D \\subseteq Z$ is invariant with sequence\n\\begin{equation*}\n\t0 \\to \\widetilde{E} \\to \\mathcal{A}_{Z,D}|_D \\to TD \\to 0,\n\\end{equation*}\nwhich in turn results in the logarithmic residue map\n\\begin{equation*}\n\t{\\rm Res}_{Z,D}\\colon \\Omega^\\bullet(Z,\\mathcal{A}_{Z,D}) \\to \\Omega^{\\bullet-1}(D;\\widetilde{E}^*).\n\\end{equation*}\nWe have $\\det(F^*) \\cong E^* \\otimes \\widetilde{E}^*$ and through this we have that ${\\rm Res}_D = {\\rm Res}_{Z,D} \\circ {\\rm Res}_Z$. Note that the line bundle $\\widetilde{E} = \\mathbb{L}_D \\to D$ is canonically trivial as in \\autoref{exa:bkersiduemaps}. The cohomological consequences of the existence of these residue maps is discussed in \\cite{KlaasseLanius18}.\n\\end{exa}\nFurther examples of residue maps for other Lie algebroids can be found in \\cite{KlaasseLanius18}.\n\\subsection{Poisson modules and modular residue maps}\n\\label{sec:poissonmodulesmodfoliation}\nIn this section we discuss modules associated to $\\mathcal{A}$-Poisson structures. For simplicity we will mainly focus on the case where $\\mathcal{A} = TX$ \\cite{GualtieriLi14,GualtieriPym13,Polishchuk97}. In essence, given a Lie algebroid $\\mathcal{A} \\to X$, $\\mathcal{A}$-Poisson modules are Lie algebroid modules for the $\\mathcal{A}$-Poisson algebroid $\\mathcal{A}^*_{\\pi_\\mathcal{A}} \\to X$ of an $\\mathcal{A}$-Poisson structure $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$. Unravelling \\autoref{defn:amodule} gives the following definition of an $\\mathcal{A}$-Poisson module.\n\\begin{defn} Let $\\pi_\\mathcal{A} \\in {\\rm Poiss}(\\mathcal{A})$. A \\emph{$\\pi_\\mathcal{A}$-module} is a bundle $E \\to X$ equipped with a flat $\\mathcal{A}$-Poisson connection, i.e.\\ a linear morphism $\\nabla\\colon \\Gamma(E) \\to \\Gamma(\\mathcal{A} \\otimes E)$ satisfying the Leibniz rule\n\t%\n\t\\begin{equation*}\n\t\\nabla(f s) = - \\pi_{\\mathcal{A}}^\\sharp(d_\\mathcal{A} f) \\otimes s + f \\nabla s,\n\t\\end{equation*}\n\tfor all $f \\in C^\\infty(X)$ and $s \\in \\Gamma(E)$, and which for all $\\alpha,\\beta \\in \\Gamma(\\mathcal{A}^*_{\\pi_{A}})$ satisfies (with $\\nabla_\\alpha s = (\\nabla s)(\\alpha)$)\n\t%\n\t\\begin{equation*}\n\t\t\\nabla_{[\\alpha,\\beta]_{\\pi_\\mathcal{A}}} = \\nabla_\\alpha \\circ \\nabla_\\beta - \\nabla_\\beta \\circ \\nabla_\\alpha.\n\t\\end{equation*}\n\t%\n\\end{defn}\nWhen $\\mathcal{A} = TX$, the bracket on $T^*_\\pi X$ is described directly by the relation $[df,dg]_\\pi = d\\{f,g\\}_\\pi$. As such, in this case flatness is given by $\\nabla_{d\\{f,g\\}_\\pi} = \\nabla_{df} \\circ \\nabla_{dg} - \\nabla_{dg} \\circ \\nabla_{df}$ as operators on $\\Gamma(E)$.\n\\begin{exa}\\label{exa:modularreppoisson} Given $\\pi \\in {\\rm Poiss}(X)$, the Lie algebroid $T^*_\\pi X$ has a representation on $\\det(T^*X)$ which is given, for $\\alpha \\in \\Gamma(T^*_\\pi X)$ and $\\mu \\in \\Gamma(\\det(T^*X))$, by\n\\begin{equation*}\n\t\\nabla_\\alpha \\mu = \\mathcal{L}_{\\pi^\\sharp(\\alpha)} \\mu + (\\pi,d\\alpha) \\mu = [\\alpha,\\mu]_\\pi - (\\pi,d\\alpha)\\mu = \\alpha \\wedge d(\\iota_\\pi \\mu).\n\\end{equation*}\nNote that this is essentially the modular representation $Q_{T^*_\\pi X} \\cong \\det(T^*_\\pi X)^2$ of $T^*_\\pi X$.\n\\end{exa}\nThe \\emph{modular class} ${\\rm Mod}(\\pi)$ of $\\pi \\in {\\rm Poiss}(X)$ is given by $2\\, {\\rm Mod}(\\pi) = {\\rm Mod}(T^*_\\pi X) = \\theta_{Q_{T^*_\\pi X}}$.\nGiven a $\\pi_{\\mathcal{A}}$-line bundle $(L,\\nabla)$, a local trivialization $s \\in \\Gamma(L)$ specifies a unique $\\mathcal{A}$-Poisson section $v \\in {\\rm Poiss}(\\pi_{\\mathcal{A}}) \\subseteq \\Gamma(\\mathcal{A})$ by $\\nabla s = v \\otimes s$. In fact, $v$ is $\\mathcal{A}$-Poisson if and only if $\\nabla$ is flat.\n\\begin{rem} For $\\pi \\in {\\rm Poiss}(X)$, in case $L = \\det(T^*X)$ with local covolume form $\\mu$, the above section $v = V_\\mu \\in \\Gamma(TX)$ is called the \\emph{$\\pi$-modular vector field} associated to $\\mu$. For general Lie algebroids $\\mathcal{A}$, the above section then gives the \\emph{$\\pi_{A}$-modular section} $V_{\\mu_\\mathcal{A}}$ associated to $\\mu_\\mathcal{A}$.\n\\end{rem}\n\\begin{exa} Given $\\pi \\in {\\rm Poiss}(X^n)$ locally expressed as $\\pi = \\sum_{ij} \\pi^{ij} \\partial_{x_i} \\wedge \\partial_{x_j}$, the modular vector field associated to $\\mu = \\partial_{x_1} \\wedge \\dots \\wedge \\partial_{x_n}$ is given by $V_\\mu = \\sum_i \\partial_k(\\pi^{ik}) \\partial_{x_i}$.\n\\end{exa}\nWe defined $\\mathcal{A}$-Poisson submanifolds as those equipped with Lie subalgebroids $(Y,\\mathcal{A}',\\pi_{\\mathcal{A}'})$ for which the inclusion is an $\\mathcal{A}$-Poisson map. We now specify to the case $\\mathcal{A} = TX$ to discuss restriction of Poisson modules. Recall that $Y \\subseteq (X,\\pi)$ is $\\pi$-Poisson if $I_Y$ is a $\\pi$-Poisson ideal.\n\\begin{defn}[\\cite{GualtieriPym13,Polishchuk97}] Let $\\pi \\in {\\rm Poiss}(\\mathcal{A})$ be given. A submanifold $Y \\subseteq X$ is a \\emph{strong $\\pi$-Poisson submanifold} if $\\mathcal{L}_V I_Y \\subseteq I_Y$ for all $V \\in {\\rm Poiss}(\\pi)$.\n\\end{defn}\nAs every $\\pi$-Hamiltonian vector field is $\\pi$-Poisson, it is clear that strong $\\pi$-Poisson submanifolds are also $\\pi$-Poisson submanifolds. In general, a \\emph{strong $\\pi$-Poisson ideal} is an ideal $I \\subseteq C^\\infty(X)$ that is preserved by all $\\pi$-Poisson vector fields. \\autoref{rem:preservingideals} shows that strong $\\pi$-Poisson ideals are closed under sums and intersections, and \\autoref{rem:preservingradical} shows the same for taking radicals. As such, we see that (strong) $\\pi$-Poisson submanifolds are closed under unions and intersections. There moreover is a strengthening of \\autoref{prop:pidegenpipoisson} (c.f.\\ \\cite{GualtieriPym13,Polishchuk97}). \n\\begin{prop}[\\cite{Polishchuk97}]\\label{prop:pidegpistrongpoisson} Let $\\pi \\in {\\rm Poiss}(X)$. Then the degeneracy ideals $I_{\\pi,2k}$ are strong $\\pi$-Poisson ideals, hence the degeneracy loci $X_{\\pi,2k}$ are strong $\\pi$-Poisson submanifolds if smooth.\n\\end{prop}\n\\bp Unravelling definitions, for $k \\geq 0$, the ideal $I_{2k,\\pi}$ is defined as the image $\\pi^k(\\Gamma(\\wedge^{2k} T^*X))$. Let $V \\in {\\rm Poiss}(\\pi)$ and $\\alpha \\in \\Gamma(\\wedge^{2k} T^*X)$. Then we readily compute that\n\\begin{equation*}\n\t\\mathcal{L}_V(\\pi^k(\\alpha)) = (\\mathcal{L}_V \\pi^k)(\\alpha) + \\pi^k(\\mathcal{L}_V \\alpha) = \\pi^k(\\mathcal{L}_V \\alpha) \\in I_{2k,\\pi}.\\qedhere\n\\end{equation*}\n\\end{proof}\nEssentially, the above follows because the degeneracy loci are defined directly in terms of the (rank of the) Poisson bivector itself: all $\\pi$-Poisson vector fields preserve all powers of $\\pi$. Turning this on its head, the above proposition says that given $\\pi \\in {\\rm Poiss}(X)$, all $\\pi$-Poisson vector fields must be tangent to all the degeneracy loci of $\\pi$.\n\nConsider a submanifold $Y \\subseteq X$ and the sequence of its conormal bundle:\n\\begin{equation*}\n\t0 \\to N^*Y \\to T^*X|_Y \\to T^*Y \\to 0.\n\\end{equation*}\nWhen $Y$ is a $\\pi$-Poisson submanifold, this becomes an extension sequence of Lie algebroids, using the Poisson algebroids $T^*X_\\pi$ and $T^*Y_{\\pi_Y}$, where $\\pi_Y \\in {\\rm Poiss}(Y)$ is the induced Poisson structure on $Y$. One naturally wonders about restriction of $\\pi$-Poisson line bundles. We have:\n\\begin{prop}[{\\cite[Lem 4.11]{GualtieriPym13}}]\\label{prop:degenlocusstrongpoisson} Let $\\pi \\in {\\rm Poiss}(X)$ and let $(L,\\nabla)$ be a $\\pi$-Poisson line bundle, and let $Y \\subseteq X$ be a strong $\\pi$-Poisson submanifold with induced Poisson structure $\\pi_Y$. Then the restriction $(L|_Y, \\nabla|_Y)$ is a $\\pi_Y$-Poisson module.\n\\end{prop}\n\\bp Let $s \\in \\Gamma(L)$ be a local trivialization. Then $\\nabla s = v \\otimes s$ for a unique $\\pi$-Poisson vector field $v \\in {\\rm Poiss}(\\pi)$. To be able to restrict $\\nabla s$ to $Y$, the image $[v \\otimes s] \\in \\Gamma(Y,{\\rm Hom}(N^*Y \\otimes L|_Y))$ is zero. As $Y$ is a strong $\\pi$-Poisson submanifold we have $\\mathcal{L}_v I_Y \\subseteq I_Y$. Thus $v$ is tangent to $Y$, so that $[v \\otimes s] \\equiv 0$. We conclude that $v \\otimes s|_Y \\in \\Gamma(Y; T^*Y \\otimes L|_Y)$ as required.\n\\end{proof}\nAs explained in \\cite{GualtieriPym13}, a $\\pi$-Poisson line bundle $(L,\\nabla)$ has several \\emph{residue maps}. Given $k \\geq 0$, with $Y := X_{\\pi,2k}$, these are sections ${\\rm Res}_k(\\nabla) \\in \\Gamma(Y,\\wedge^{2k+1} TY)$. They are locally given using a trivialization $s \\in \\Gamma(L)$, so that $\\nabla s = v \\otimes s$ for $v$ a local $\\pi$-Poisson vector field, by the expression\n\\begin{equation*}\n\t{\\rm Res}_k(\\nabla) = v \\wedge \\pi^k_Y.\n\\end{equation*}\nInvariantly, they are obtained by the composition\n\\begin{equation*}\n\t(\\cdot \\wedge \\sigma^k) \\circ \\nabla\\colon \\Gamma(L) \\to \\Gamma(TX \\otimes L) \\to \\Gamma(\\wedge^{2k+1}TX \\otimes L),\n\\end{equation*}\nwhich is then restricted to $Y$ using \\autoref{prop:degenlocusstrongpoisson} to give a morphism\n\\begin{equation*}\n\t\\Gamma(Y, L|_Y) \\to \\Gamma(Y, \\wedge^{2k+1}TY \\otimes L|_Y),\n\\end{equation*}\nfrom which the multiderivation ${\\rm Res}_k(\\nabla) \\in \\Gamma(Y, \\wedge^{2k+1} TY)$ is extracted. During this process it is used that $\\pi^{k+1}|_Y \\equiv 0$ because $Y$ is the $2k$th-degeneracy locus of $\\pi$. Applying the above to $L = \\det(T^*X)$ we obtain the \\emph{$k$th modular residue} ${\\rm Res}_{k,{\\rm mod}}(\\pi)$.\n\nIn \\autoref{sec:residuesympla} we briefly discuss these residue maps and their relation to those in \\autoref{sec:residuemaps} in the context of Poisson structures of nondegenerate $\\mathcal{A}$-type for a Lie algebroid $\\mathcal{A} \\to X$.\n\\subsection{The modular foliation}\n\\label{sec:modularfoliation}\nIn this section we discuss the modular foliation of a Poisson manifold, which was introduced in \\cite{GualtieriPym13}. Our main goal here is to prove \\autoref{thm:modfoliation} which discusses how the modular foliation interacts with the lifting procedure of \\autoref{sec:liftingpoisson}.\n\nLet $\\pi \\in {\\rm Poiss}(X)$ be given and let $(L,\\nabla) \\to X$ be a $\\pi$-Poisson line bundle. Then ${\\rm At}(L) \\to X$, its \\emph{Atiyah algebroid}, is a transitive Lie algebroid. There is a natural section $\\sigma_\\nabla \\in \\Gamma(\\wedge^2 {\\rm At}(L))$, which is in fact an ${\\rm At}(L)$-Poisson structure (\\cite[Corollary 5.3]{Polishchuk97}). It can be viewed as the natural Poisson structure on ${\\rm Tot}(L)$ induced by $\\pi$ and $\\nabla$. In our language, $\\sigma_\\nabla$ is a Poisson lift of the underlying Poisson structure $\\pi$. Due to this, $({\\rm At}(L), {\\rm At}^*(L), \\sigma_\\nabla)$ is a triangular Lie bialgebroid (c.f.\\ \\autoref{rem:apoissontriangbialgebroid}). In \\autoref{sec:poissondivtype} we studied cases where these were of divisor-type, but here instead the Lie algebroid ${\\rm At}(L)$ is transitive. We have the associated module $\\mathcal{F}_{\\sigma_\\nabla} = \\sigma_\\nabla^\\sharp(\\Gamma({\\rm At}(L)^*) \\subseteq \\Gamma({\\rm At}(L))$, which can be pushed down using $\\rho\\colon {\\rm At}(L) \\to TX$.\n\\begin{defn}[\\cite{GualtieriPym13}] Let $\\pi \\in {\\rm Poiss}(X)$ and $(L,\\nabla) \\to X$ a $\\pi$-Poisson line bundle. Then $\\mathcal{F}_{\\pi,(L,\\nabla)} := \\rho(\\mathcal{F}_{\\sigma_\\nabla}) \\subseteq \\Gamma(TX)$ is the singular foliation associated to $(L,\\nabla)$.\n\\end{defn}\nIt is immediate that the module $\\mathcal{F}_{\\pi,(L,\\nabla)}$ is involutive as $\\rho \\circ \\sigma_\\nabla^\\sharp$ is a Lie algebroid morphism.\n\\begin{defn}[\\cite{GualtieriPym13}] Let $\\pi \\in {\\rm Poiss}(X)$. Then the \\emph{modular foliation} $\\mathcal{F}_{\\pi,{\\rm mod}}$ is the singular foliation associated to the modular representation $L = \\det(T^*X)$ of $\\pi$ (see \\autoref{exa:modularreppoisson}).\n\\end{defn}\nLet us describe the modular foliation locally, as in \\cite{PymSchedler18}. Let $\\pi \\in {\\rm Poiss}(X)$ and let $\\mu$ be a local volume form. Then the associated modular vector field $V_{\\pi,\\mu}$ of $\\pi$ is characterized by\n\\begin{equation*}\n\t\\nabla(\\mu) = V_{\\pi,\\mu} \\otimes \\mu, \\qquad \\text{or} \\qquad \\mathcal{L}_{\\pi^\\sharp(df)} \\mu = (\\mathcal{L}_{V_{\\pi,\\mu}} f) \\cdot \\mu.\n\\end{equation*}\nMoreover, it is immediate that $V_{\\pi,\\mu} \\in {\\rm Poiss}(\\pi)$. Consequently, $V_{\\pi,\\mu}$ is always tangent to the degeneracy loci of $\\pi$ by \\autoref{prop:pidegenpipoisson}. The same computation showing that the characteristic class $\\theta_L$ is independent of local generators shows that if we change volume element to $\\mu' = f \\mu$ for some nonvanishing function $f$, we obtain that $V_{\\pi,\\mu'} = V_{\\pi,\\mu} - V_{\\log f}$, i.e.\\ the associated modular vector fields differ by a Hamiltonian vector field. As a result we have\n\\begin{equation*}\n\t\\mathcal{F}_{\\pi,{\\rm mod}} = \\mathcal{F}_\\pi \\cup \\{V_{\\pi,\\mu} \\, | \\, \\mu \\in {\\rm Vol}_{\\rm loc}(X)\\}.\n\\end{equation*}\nThe involutivity of $\\mathcal{F}_{\\pi,{\\rm mod}}$ locally becomes the computation that, given $f \\in C^\\infty(X)$, we have:\n\\begin{equation*}\n\t[V_{\\pi,\\mu}, V_f] = \\mathcal{L}_{V_{\\pi,\\mu}}(\\pi^\\sharp(df)) = \\pi^\\sharp(d \\mathcal{L}_{V_{\\pi,\\mu}} f) = V_{\\mathcal{L}_{V_{\\pi,\\mu}} f}.\n\\end{equation*}\nWe see that $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}}$. The leaves of the former are the $\\pi$-symplectic leaves, which are thus necessarily even-dimensional. The leaves of the modular folation come in two flavors:\n\\begin{itemize}\n\t\\item The even-dimensional leaves, where $V_{\\pi,{\\rm mod}}$ is tangent to the symplectic leaves;\n\t\\item The odd-dimensional leaves, where $V_{\\pi,{\\rm mod}}$ is transverse to the symplectic leaves.\n\\end{itemize}\nLet us record the consequence of \\autoref{prop:pidegpistrongpoisson} regarding strong Poisson submanifolds.\n\\begin{prop}\\label{prop:modvfielddegenloci} Let $\\pi \\in {\\rm Poiss}(X)$. Then any local modular vector field for $\\pi$ is tangent to all strong Poisson submanifolds of $\\pi$, hence in particular to all of its degeneracy loci.\n\\end{prop}\nThe following (combined with \\autoref{prop:atypepoissonsubmfd} showing $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_\\mathcal{A}$) is \\autoref{thm:intromodfoliation}.\n\\begin{thm}\\label{thm:modfoliation} Let $\\pi \\in {\\rm Poiss}(X)$ be of $\\mathcal{A}$-type. Then $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\mathcal{F}_\\mathcal{A}$.\n\\end{thm}\n\\bp As noted, \\autoref{prop:atypepoissonsubmfd} shows that $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_\\mathcal{A}$, while we always have $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}}$. We are left with showing that all local modular vector fields are contained in $\\mathcal{F}_\\mathcal{A}$. Let $\\pi_{A} \\in {\\rm Poiss}(\\mathcal{A})$ be an $\\mathcal{A}$-lift of $\\pi$. Given local trivializing sections $\\nu \\in \\Gamma(\\det(\\mathcal{A}^*))$ and $\\mu \\in \\Gamma(\\det(T^*X))$, note that $\\nu \\otimes \\mu \\in  \\Gamma(\\det(\\mathcal{A}^*) \\otimes \\det(T^*X))$ can serve as a nonvanishing section for the modular representation $Q_{\\mathcal{A}^*_{\\pi_\\mathcal{A}}}$ of the Poisson algebroid $\\mathcal{A}^*_{\\pi_\\mathcal{A}}$. These now give rise to the modular section $v_{\\nu \\otimes \\mu} \\in \\Gamma(\\mathcal{A})$, but also to the $\\pi_\\mathcal{A}$-modular section $V_{\\pi_\\mathcal{A},\\nu} \\in \\Gamma(\\mathcal{A})$ and the $\\pi$-modular vector field $V_{\\pi,\\mu} \\in \\Gamma(TX)$. By \\cite[Equation (64)]{KosmannSchwarzbach00}, these are related by the equality\n\\begin{equation*}\n\t\\rho_\\mathcal{A}(v_{\\nu \\otimes \\mu} - V_{\\pi_\\mathcal{A},\\nu}) = V_{\\pi,\\mu}.\n\\end{equation*}\nThis shows that $V_{\\pi,\\mu} \\in \\mathcal{F}_\\mathcal{A}$, from which $\\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\mathcal{F}_\\mathcal{A}$ follows.\n\\end{proof}\nIn \\autoref{cor:poissonhamiltonian} we showed that if $\\pi$ is of $I$-divisor-type, then any $\\pi$-Poisson vector field admits a $TX_I$-lift. This shows that $\\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\Gamma(TX)_I = \\rho_{TX_I}(\\Gamma(TX_I))$ because any $\\pi$-modular vector field is $\\pi$-Poisson. Hence, it is not always necessary that $\\pi$ is of $\\mathcal{A}$-type.\n\\begin{rem} It is an interesting question to ask when the modular foliation $\\mathcal{F}_{\\pi,{\\rm mod}}$ is projective. Note that $\\mathcal{F}_\\pi$ is projective if and only if $\\pi$ is of $m$-divisor-type, i.e.\\ is almost-regular (\\autoref{cor:divtypealmostreg}). How is projectivity of $\\mathcal{F}_{\\pi,{\\rm mod}}$ related to projectivity of $\\mathcal{F}_\\pi$?\n\\end{rem}\n\\subsubsection{Relations between modular classes}\nThere are extensions of the above concepts to $\\mathcal{A}$-Poisson structures, using ideas contained in \\cite{CaseiroFernandes13,KosmannSchwarzbachLaurentGengouxWeinstein08,KosmannSchwarzbach00, KosmannSchwarzbach08,KosmannSchwarzbachLaurentGengoux05,KosmannSchwarzbachWeinstein05}. Recall from \\autoref{sec:aconnareps} that any Lie algebroid $\\mathcal{A} \\to X$ has a modular class ${\\rm Mod}(\\mathcal{A})$. As an $\\mathcal{A}$-Poisson structure turns $\\mathcal{A}^*$ into a Lie algebroid (\\autoref{defn:apoissonalgebroid}), there are various modular classes around. In particular, the modular class ${\\rm Mod}(\\pi_{\\mathcal{A}})$ of $\\pi_{\\mathcal{A}}$ is related to ${\\rm Mod}(\\pi)$ through the modular class of $\\mathcal{A}$: an $\\mathcal{A}$-Poisson structure $\\pi_{\\mathcal{A}}$ comes equipped with:\n\\begin{itemize}\n\t\\item A Lie algebroid morphism $\\pi_{\\mathcal{A}}^\\sharp\\colon \\mathcal{A}^*_{\\pi_{\\mathcal{A}}} \\to \\mathcal{A}$;\n\t\\item A Lie algebroid morphism $\\rho_{A}\\colon \\mathcal{A} \\to TX$;\n\t\\item A Lie algebroid morphism $\\pi^\\sharp\\colon T^*X_\\pi \\to TX$;\n\t\\item A Lie algebroid comorphism $\\rho_{A}^*\\colon \\mathcal{A}^*_{\\pi_{\\mathcal{A}}} \\dashrightarrow T^*X_\\pi$.\n\\end{itemize}\nThese are of course related by the lifting relation $\\pi^\\sharp = \\rho_{A} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{A}^*$. As $\\rho_{A}^*$ is base-preserving, it can also be considered as a Lie algebroid morphism from $T^*X_\\pi$ to $\\mathcal{A}^*_{\\pi_\\mathcal{A}}$. Each of these four Lie algebroids has a modular class, and each morphism has a relative modular class \\cite{KosmannSchwarzbachWeinstein05}. These are related to (or define) the modular classes ${\\rm Mod}(\\pi_\\mathcal{A})$ and ${\\rm Mod}(\\pi)$ as follows:\n\\begin{itemize}\n\t\\item $2\\, {\\rm Mod}(\\pi_{\\mathcal{A}}) = {\\rm Mod}^{\\pi_{\\mathcal{A}}^\\sharp}(\\mathcal{A}^*_{\\pi_\\mathcal{A}},\\mathcal{A}) = {\\rm Mod}(\\mathcal{A}^*_{\\pi_\\mathcal{A}}) - (\\pi_\\mathcal{A}^\\sharp)^*({\\rm Mod}(\\mathcal{A}))$;\n\t\\item $2\\, {\\rm Mod}(\\pi) = {\\rm Mod}^{\\pi^\\sharp}(T^*X_\\pi, TX) = {\\rm Mod}(T^*X_\\pi) - (\\pi^\\sharp)^*({\\rm Mod}(TX)) = {\\rm Mod}(T^*X_\\pi)$;\n\t\\item ${\\rm Mod}^{\\rho_{A}}(\\mathcal{A},TX) = {\\rm Mod}(\\mathcal{A}) - \\rho_{A}^*({\\rm Mod}(TX)) = {\\rm Mod}(\\mathcal{A})$, because ${\\rm Mod}(TX) = 0$;\n\t\\item ${\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) = {\\rm Mod}(T^*X_\\pi) - \\rho_{A}({\\rm Mod}(\\mathcal{A}^*_{\\pi_{\\mathcal{A}}}))$, as $(\\rho_\\mathcal{A}^*)^* = \\rho_\\mathcal{A}$.\n\\end{itemize}\nThe equality $\\pi^\\sharp = \\rho_{A} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{A}^*$ then implies the following relations:\n\t%\n\t\\begin{align*}\n\t\t2\\, {\\rm Mod}(\\pi) &= {\\rm Mod}^{\\pi^\\sharp}(T^*X_\\pi, TX) = {\\rm Mod}^{\\rho_{A} \\circ \\pi_\\mathcal{A}^\\sharp \\circ \\rho_{A}^*}(T^*X_\\pi, TX)\\\\\n\t\t&= {\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) + \\rho_{A}({\\rm Mod}^{\\rho_\\mathcal{A} \\circ \\pi_\\mathcal{A}^\\sharp}(\\mathcal{A}^*_{\\pi_\\mathcal{A}}, TX))\\\\\n\t\t&= {\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) + \\rho_{A}({\\rm Mod}^{\\pi_\\mathcal{A}^\\sharp}(\\mathcal{A}_{\\pi_\\mathcal{A}}^*,\\mathcal{A}) + (\\pi_{A}^\\sharp)^*({\\rm Mod}^{\\rho_\\mathcal{A}}(\\mathcal{A},TX)))\\\\\n\t\t&= {\\rm Mod}^{\\rho_{A}^*}(T^*X_\\pi, \\mathcal{A}^*_{\\pi_\\mathcal{A}}) + \\rho_{A}(2\\,{\\rm Mod}(\\pi_\\mathcal{A}) + (\\pi_{A}^\\sharp)^*({\\rm Mod}(\\mathcal{A}))).\n\t\\end{align*}\nThis type of description can possibly be used in the discussion of liftability of unimodular Poisson structures to unimodular Lie algebroids.\n\\subsubsection{Examples of modular foliations}\nWe finish by describing several examples of modular foliations for Poisson structures of divisor-type, starting locally in dimension two.\n\\begin{exa} As in \\cite[Example 3.1]{PymSchedler18}, given $\\pi = f \\partial_x \\wedge \\partial_y$ on $(\\mathbb{R}^2,(x,y))$, its modular vector field associated to $\\mu = \\partial_x \\wedge \\partial_y$ is given by $V_{\\pi,\\mu} = (\\partial_x f) \\partial_y - (\\partial_y f) \\partial_x$. The symplectic foliation is given by the subset where $f$ is nonvanishing, and the points where $f$ vanishes. The modular vector field is tangent to the zero set of $f$, and vanishes exactly at its critical points.\n\\end{exa}\n\\begin{exa}[Nondegenerate] Let $\\pi \\in {\\rm Poiss}(X)$ be nondegenerate. Then $X$ is a symplectic leaf (or: its connected components). This shows that $\\mathcal{F}_\\pi = \\Gamma(TX)$, as is also immediate as $\\pi^\\sharp\\colon T^*X \\to TX$ is an isomorphism. The only degeneracy locus of $\\pi$ is $X$ itself, and any modular vector field must be tangent to it by \\autoref{prop:modvfielddegenloci}. Thus $\\mathcal{F}_{\\pi,{\\rm mod}} = \\mathcal{F}_{\\pi} = \\Gamma(TX)$. In fact, nondegenerate Poisson manifolds are unimodular, so that their modular class vanishes and there exists a global volume form (the Liouville form) which is preserved by Hamiltonian flows. For this volume form the modular vector field vanishes, which again shows that $\\mathcal{F}_\\pi = \\mathcal{F}_{\\pi, {\\rm mod}}$.\n\\end{exa}\n\\begin{exa}[Log-Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be a log-Poisson structure. Then $X$ and $Z$ are the only degeneraci loci of $\\pi$. From this it is immediate that any local modular vector field must be tangent to $Z$. However, in fact any local modular vector field is transverse to the symplectic leaves of the induced Poisson structure $\\pi_Z$ \\cite{GualtieriLi14,GuilleminMirandaPires14}. As a consequence we have that $\\mathcal{F}_{\\pi_Z,{\\rm mod}} = \\Gamma(TZ)$, while $\\mathcal{F}_{\\pi,{\\rm mod}} = \\Gamma(TX,TZ) = \\Gamma(\\mathcal{A}_Z)$. By \\autoref{exa:logpoissonlift} we know that $\\pi$ is of $\\mathcal{A}_Z$-type, so that \\autoref{thm:modfoliation} tells us that $\\mathcal{F}_\\pi \\subseteq \\mathcal{F}_{\\pi,{\\rm mod}} \\subseteq \\mathcal{F}_{\\mathcal{A}_Z} = \\Gamma(\\mathcal{A}_Z)$. We see that the second inclusion is an equality, and that $\\mathcal{F}_\\pi$ and $\\mathcal{F}_{\\pi, {\\rm mod}}$ are both projective. The symplectic foliation of $\\pi_Z$ is regular of corank-1, with tangencies $TF \\subseteq TZ$, so that $\\mathcal{F}_\\pi = \\Gamma(TX,TF)$.\n\\end{exa}\n\\begin{exa}[$b^k$-Poisson] Let $\\pi \\in {\\rm Poiss}(X)$ be a $b^k$-Poisson structure. Then the behavior of $\\pi$ is similar for all $k \\geq 1$, and in particular to log-Poisson structures (where $k = 1$). Again the only degeneraci loci of $\\pi$ are $X$ and $Z$, but if $k > 1$, then any modular vector field is tangent to the symplectic leaves of $\\pi_Z$. If locally $\\pi = z^k \\partial_z \\wedge \\partial_{x_1} + \\omega_0^{-1}$ as in \\autoref{exa:bkpoissondarboux}, then $V_{\\pi,\\mu} = k z^{k-1} \\partial_{x_1}$ with respect to $\\mu = dz \\wedge dx_i$. In terms of this local description, we then have\n\\begin{equation*}\n\t\\mathcal{F}_\\pi = \\langle z^k \\partial_z, z^k \\partial_{x_1} \\rangle \\oplus \\Gamma(TF), \\quad \\mathcal{F}_{\\pi,{\\rm mod}} = \\langle z^k \\partial_z, z^{k-1} \\partial_{x_1} \\rangle \\oplus \\Gamma(TF), \\quad \\mathcal{F}_{\\mathcal{A}_Z^k} = \\langle z^k \\partial_z, \\partial_{x_1} \\rangle \\oplus \\Gamma(TF),\n\\end{equation*}\n where $TF \\subseteq TX$ are the tangencies to the corank-one symplectic foliation of $\\pi_Z \\in {\\rm Poiss}(X)$.\n\\end{exa}\nThe modular foliations of elliptic and elliptic-log Poisson manifolds are described in \\cite{KlaasseLanius18,KlaasseLi18}.\n\\begin{exa}[$m$-Log-Poisson]\\label{exa:modfolmlog} Let $\\pi \\in {\\rm Poiss}(X)$ be $m$-log-Poisson with associated regular distribution $D$. Then the degeneracy loci of $\\pi$ are $X$ and $Z$, so that any local modular vector field must be tangent to $Z$. However, as $\\pi$ is of $\\mathcal{D}$-type by \\autoref{prop:mdivtypelift}, it must further be tangent to $D$ due to \\autoref{thm:modfoliation}. This suggests that the modular vector fields can detect how the orbits of $\\mathcal{D}$ meet $Z$. In the local examples of \\autoref{rem:mlogpoissonlocal} on $(\\mathbb{R}^3,(x,y,z))$, with respect to the volume form $\\mu = dx \\wedge dy \\wedge dz$, the modular vector fields of $\\pi_1$, $\\pi_2$ and $\\pi_3$ are:\n\\begin{equation*}\n\tV_{\\pi_1,\\mu} = \\partial_y, \\quad V_{\\pi_2,\\mu} = 0, \\quad V_{\\pi_3,\\mu} = - 2x \\partial_y.\n\\end{equation*}\nFrom this we see that the associated modules $\\mathcal{F}_{\\pi_i}$ and $\\mathcal{F}_{\\pi_i, {\\rm mod}}$ are given by\n\\begin{align*}\n\t&\\mathcal{F}_{\\pi_1} = \\langle x \\partial_x, x \\partial_y \\rangle, \\quad \\mathcal{F}_{\\pi_2} = \\langle z \\partial_x, z \\partial_y \\rangle, \\quad \\mathcal{F}_{\\pi_3} = \\langle (z-x^2) \\partial_x, (z-x^2)\\partial_y \\rangle,\\\\\n\t&\\mathcal{F}_{\\pi_1,{\\rm mod}} = \\langle x \\partial_x, \\partial_y \\rangle, \\quad \\mathcal{F}_{\\pi_2,{\\rm mod}} = \\langle z \\partial_x, z \\partial_y \\rangle \\quad \\mathcal{F}_{\\pi_3,{\\rm mod}} = \\langle (z-x^2) \\partial_x, z \\partial_y, x \\partial_y. \\rangle\n\\end{align*}\n\\end{exa}\nIn general, the modular foliation of an almost-regular Poisson manifold behaves as follows: in \\autoref{sec:liftingalmostreg} we saw that if $\\pi \\in {\\rm Poiss}(X)$ is of $m$-$I$-divisor-type with associated involutive distribution $D = D_\\pi$, then $\\pi$ is of both $D$- and $TX_I$-type (assuming projectivity of $I$). Then, \\autoref{thm:modfoliation} implies that $\\mathcal{F}_{\\pi, {\\rm mod}} \\subseteq \\Gamma(\\mathcal{D}_\\pi)$ and $\\mathcal{F}_{\\pi, {\\rm mod}} \\subseteq \\Gamma(TX_I) = \\Gamma(TX)_I$. In particular any modular vector field must be tangent to $D$ and to the subset $Z_I \\subseteq X$ (if it is smooth).\n\\subsection{Residues for symplectic Lie algebroids}\n\\label{sec:residuesympla}\nIn this section we discuss some of the consequences of the fact that symplectic Lie algebroids $\\mathcal{A}$ have residue maps, which can be applied to an $\\mathcal{A}$-symplectic form $\\omega_{\\mathcal{A}}$ or its powers. We recall again that residue maps, while often chain maps, typically do not respect the ring structure coming from the wedge product. We will discuss this process in more detail in \\cite{Klaasse18} in the context of Dirac geometry, yet show here what happens in the case of $b^k$-Poisson or elliptic Poisson structures.\n\nWe will need the following definitions (\\cite[Definitions 3.2.20 and 3.2.21]{OsornoTorres15}).\n\\begin{defn} Let $M^m$ be a manifold equipped with a $(m-2n = 2\\ell)$-regular foliation $\\mathcal{F}$.\n\t%\n\t\\begin{itemize}\n\t\\item an \\emph{$n$-cosymplectic tuple} on $M$ is a tuple $(\\alpha_1, \\dots, \\alpha_n, \\beta)$ of forms, where $\\beta \\in \\Omega^2_{\\rm cl}(M)$ has constant rank $2\\ell$, $\\alpha_i \\in \\Omega^1_{\\rm cl}(M)$ for all $i$, and $\\alpha_1 \\wedge \\dots \\wedge \\alpha_n \\wedge \\beta^{\\ell} \\neq 0$;\n\t\\item an \\emph{$n$-Poisson tuple} on $M$ is a tuple $(V_1, \\dots V_n, \\pi)$, where $\\pi \\in {\\rm Poiss}(M)$ is regular of rank $2\\ell$, $V_i \\in \\Gamma(TM)$ are pairwise commuting $\\pi$-Poisson vector fields, and $V_1 \\wedge \\dots V_n \\wedge \\pi^\\ell \\neq 0$.\n\t\\end{itemize}\n\t%\n\tWe say the tuple is \\emph{adapted to $\\mathcal{F}$} if $T\\mathcal{F}$ is spanned by (the kernel of) the $v_i$ (respectively $\\alpha_i$).\n\\end{defn}\nWhen $n=1$ the above is simply referred to as a \\emph{cosymplectic} or \\emph{corank-$1$ Poisson tuple}. The following correspondence is in full analogy with the case $n=1$ contained in \\cite{GuilleminMirandaPires11}.\n\\begin{lem}[{\\cite[Lemma 3.2.29]{OsornoTorres15}}]\\label{lem:oneonecorrespondence} Let $(M,\\mathcal{F})$ be a manifold with a corank-$2n$ foliation. There is a one-to-one correspondence between $\\mathcal{F}$-adapted $n$-cosymplectic and $n$-Poisson tuples.\n\\end{lem}\nWe wish to reinterpet these structures from a Dirac geometric point of view. Recall that a \\emph{Dirac structure} $E \\subseteq \\mathbb{T}M$ is an involutive Lagrangian subbundle of the standard Courant algebroid structure on $\\mathbb{T}M = TM \\oplus T^*M$. A Dirac structure can equivalently be described by a \\emph{pure spinor line}, as follows (see \\cite{AlekseevBursztynMeinrenken09,Gualtieri11}). Sections $v = V + \\xi \\in \\Gamma(\\mathbb{T}M)$ act on differential forms via Clifford multiplication, given by $v \\cdot \\rho = \\iota_V \\rho + \\xi \\wedge \\rho$ for $\\rho \\in \\Omega^\\bullet(M)$. With this in mind, a pure spinor line is a line bundle $K_E \\subseteq \\wedge^\\bullet T^*M$ that is pointwise generated by a pure spinor $\\rho = e^B \\wedge \\Omega$ with $B$ a two-form and $\\Omega$ a decomposable $k$-form, such that for any nonzero local section $\\rho \\in \\Gamma(K_E)$ we have $d \\rho = v \\cdot \\rho$ for some $v \\in \\Gamma(\\mathbb{T}M)$. The correspondence  is then given by the fact that $E = {\\rm Ann}(K_E)$ is the annihilator of $K_E$ under the Clifford action.\n\\begin{prop} Let $M^m$ be a manifold. Then an $n$-cosymplectic tuple $(\\alpha_1,\\dots,\\alpha_n,\\beta)$ defines a global pure spinor $\\rho := e^\\beta \\wedge (\\alpha_1 \\wedge \\dots \\wedge \\alpha_n)$ for which $\\rho_m \\neq 0 \\in \\Gamma(\\wedge^m T^*M)$. \n\\end{prop}\nWe call the associated pure spinor line $K_\\rho = \\langle \\rho \\rangle$ an \\emph{$n$-cosymplectic structure} on $M$.\n\\bp This is almost immediate. Note that $\\rho_m = \\alpha_1 \\wedge \\dots \\wedge \\alpha_n \\wedge \\beta^\\ell$, which is nonvanishing. Letting $\\Omega := \\alpha_1 \\wedge \\dots \\wedge \\alpha_n$, note that because $\\beta$ and each $\\alpha_i$ is closed we have\n\\begin{equation*}\n\td \\rho = d(e^\\beta \\wedge \\Omega) = d\\beta \\wedge \\rho + e^\\beta \\wedge d\\Omega = 0 = 0 \\cdot \\rho.\n\\end{equation*}\nThe Dirac structure associated to $K_\\rho = \\langle \\rho \\rangle$ is given by $e^\\beta( T\\mathcal{F}) = \\{v + \\beta^\\flat(v) \\, | \\, v \\in \\Gamma(T\\mathcal{F})\\}$.\n\\end{proof}\nFrom the Dirac perspective, the intrinsic object is the pure spinor line $K_\\rho$, instead of $\\rho$ itself. However, $n$-cosymplectic structures have a global generating pure spinor.  In general, this still means that the one-forms $\\alpha_i$ need not exist separately, but only their associated volume form $\\alpha_1 \\wedge \\dots \\wedge \\alpha_n \\in \\Gamma(\\det(T\\mathcal{F})^*)$ (making $\\mathcal{F}$ coorientable). Their separate existence distinguishes $n$-cosymplectic structures from other symplectic foliations (such foliations are called \\emph{unimodular}). Similarly, the two-form $\\beta$ should be viewed as an extension to $M$ of a symplectic form on $\\mathcal{F}$. However, $n$-cosymplectic structures are such that the symplectic form on $\\mathcal{F}$ admits a closed global extension (such symplectic foliations are called \\emph{strong}). Thus, an $n$-cosymplectic structure determines a coorientable and unimodular corank-$2n$ strong symplectic foliation, whose associated pure spinor line has a global generating pure spinor.\n\nWe now discuss how these structures arise in the context of log-Poisson structures, and of elliptic Poisson structures with zero elliptic residue. To start, recall that log-Poisson structures admit nondegenerate lifts to the log-tangent bundles that they define (\\autoref{exa:logpoissonlift}). In other words, $\\pi \\in {\\rm Poiss}(X)$ of $I_Z$-divisor-type has an associated dual $\\mathcal{A}_Z$-symplectic structure $\\omega$.\n\\begin{prop}\\label{prop:logcosymplectic} Let $(X^{2n},Z,\\pi,\\omega)$ be a log-Poisson manifold. Then\n\\begin{equation*}\n\t\\rho := {\\rm Res}_Z(e^\\omega) = {\\rm Res}_Z\\left(\\sum_{k=1}^n \\frac{1}{k!}  \\omega^k\\right) \\in \\Omega^{\\bullet}_{\\rm cl}(Z)\n\\end{equation*}\nforms a $1$-cosymplectic global pure spinor on $Z$, hence defines a $1$-cosymplectic structure. In particular, $Z$ inherits a unimodular corank-$1$ symplectic foliation.\n\\end{prop}\nWe are asserting here that the residue map should be applied to the pure spinor $e^\\omega$ associated to the $\\mathcal{A}_Z$-Dirac structure defined by $\\omega$ (see \\cite{Klaasse18} for more discussion on this process).\n\\bp Because ${\\rm Res}_Z$ is a cochain morphism (due to \\autoref{thm:residues}), we have\n\\begin{equation*}\n\td\\rho = d \\, {\\rm Res}_Z\\left(\\sum_{k=1}^n \\frac{1}{k!} \\omega^k\\right) = {\\rm Res}_Z\\left(\\sum_{k=1}^n \\frac{1}{k!} d (\\omega^k)\\right) = 0.\n\\end{equation*}\n Let $z$ be a local defining function for $\\iota_Z\\colon Z \\hookrightarrow X$, so that locally $\\omega = d\\log z \\wedge \\widetilde{\\alpha} + \\widetilde{\\beta}$. Then \n\\begin{equation*}\n\t\\omega^k = k (d\\log z \\wedge \\widetilde{\\alpha} + \\widetilde{\\beta}) \\wedge \\widetilde{\\beta}^{k-1}, \\qquad 1 \\leq k \\leq n.\n\\end{equation*}\nThus ${\\rm Res}_Z(\\omega^k) = k \\alpha \\wedge \\beta^{k-1} \\neq 0$ for $\\alpha := {\\rm Res}_Z(\\omega) = \\iota_Z^*(\\widetilde{\\alpha})$ and $\\beta := \\iota_Z^*(\\widetilde{\\beta})$ (see \\autoref{exa:bkersiduemaps}). Both $\\rho$ and $\\alpha$ are invariantly defined closed forms on $Z$, and $\\rho$ is a global pure spinor. We see that $\\rho$ is locally given by\n\\begin{equation*}\n\t\\rho = \\alpha \\wedge e^\\beta.\n\\end{equation*}\nMoreover, $\\rho_{2n-1} = \\frac{1}{(n-1)!} \\alpha \\wedge \\beta^{n-1}$, which is nonzero due to nondegeneracy of $\\omega$ (i.e.\\ $\\omega^n \\neq 0$).\n\\end{proof}\n\\begin{rem} \\autoref{prop:logcosymplectic} is the intrinsic reformulation of \\cite[Proposition 10]{GuilleminMirandaPires14}, that log-Poisson structures carry an equivalence class of cosymplectic structures $(\\alpha,\\beta)$, where $(\\alpha,\\beta) \\sim (\\alpha',\\beta')$ if $\\alpha = \\alpha'$ and $\\beta' = \\beta + \\alpha \\wedge df$ for $f \\in C^\\infty(Z)$. There they note that the one-form $\\alpha$ is determined invariantly, and here this is immediate because it arises from the logarithmic residue map ${\\rm Res}_Z$. Any choice of (local) defining function for $Z$ picks out a representative for $\\beta$, but the associated pure spinor $\\rho$ is the invariant object, globally on $Z$. This is again because it is obtained by (repeated) application of the logarithmic residue.\n\\end{rem}\nBy repeating the proof of \\autoref{prop:logcosymplectic} essentially verbatim, we obtain the following invariant statement about the induced geometry on the degeneracy loci of $b^k$-Poisson structures, which lead to symplectic Lie algebroids by \\autoref{exa:bkpoissonlift} (compare this with \\cite{Scott16,GuilleminMirandaWeitsman17}).\n\\begin{prop}\\label{prop:bkpoissoncosymplectic} Let $(X,Z,\\pi)$ be a $b^k$-Poisson manifold. Then $Z$ carries a $1$-cosymplectic global pure spinor, so that it inherits a $1$-cosymplectic structure.\n\\end{prop}\nThere is a similar result for elliptic Poisson manifolds, as follows. Due to \\autoref{exa:ellpoissonlift}, an elliptic Poisson structure $\\pi \\in {\\rm Poiss}(X)$ (with divisor ideal $I_{|D|}$) is of nondegenerate $\\mathcal{A}_{|D|}$-type, with dual $\\mathcal{A}_{|D|}$-symplectic structure $\\omega$. We will focus on the case where ${\\rm Res}_q(\\omega) = 0$, which are those with zero elliptic residue, or \\emph{zero elliptic Poisson structures}. We will moreover require that $D$ is cooriented (which is immediate if $\\pi$ stems from a stable generalized complex structure \\cite{CavalcantiGualtieri18}). As mentioned in \\autoref{exa:elltangentresidues}, this trivializes the isotropy line $\\mathfrak{k} \\to D$, so that the residue one-forms ${\\rm Res}_r(\\omega) \\in \\Omega^1(D)$ and ${\\rm Res}_\\theta(\\omega) \\in \\Omega^1(D)$ are both closed.\n\\begin{prop}\\label{prop:elliptic2cosymplectic} Let $(X^{2n},|D|,\\pi,\\omega)$ be a cooriented zero elliptic Poisson manifold. Then\n\\begin{equation*}\n\t\\rho := {\\rm Res}_q(e^\\omega) = -{\\rm Res}_r(\\omega) \\wedge {\\rm Res}_{\\theta}(\\omega) + {\\rm Res}_q \\left(\\sum_{k=3}^{n} \\frac1{k!} \\omega^k \\right) \\in \\Omega^\\bullet_{\\rm cl}(D)\n\\end{equation*}\nforms a $2$-cosymplectic global pure spinor on $D$, hence defines a $2$-cosymplectic structure. In particular, $D$ inherits a unimodular corank-$2$ symplectic foliation.\n\\end{prop}\nThere are extensions of this statement when $D$ is not cooriented (see \\cite{KlaasseLanius18,KlaasseLi18}). Moreover, this result is known in the context of stable generalized complex geometry due to \\cite{BaileyCavalcantiGualtieri17,CavalcantiGualtieri18}. There are three residue maps, namely ${\\rm Res}_r$, ${\\rm Res}_\\theta$ and ${\\rm Res}_q$, so that the pure spinor above differs slightly from \\autoref{prop:logcosymplectic}. We have that ${\\rm Res}_r(\\omega) \\wedge {\\rm Res}_\\theta(\\omega) = -{\\rm Res}_q(\\omega^2)$. As for \\autoref{prop:logcosymplectic} we apply ${\\rm Res}_q$ to the pure spinor $e^\\omega$ defining the $\\mathcal{A}_{|D|}$-symplectic structure.\n\\bp Let $(r,\\theta)$ be local normal polar coordinates to $\\iota_D\\colon D \\hookrightarrow X$ in which $I_{|D|} = \\langle r^2 \\rangle$. Then locally $\\omega = d\\log r \\wedge \\widetilde{\\alpha}_1 + d\\theta \\wedge \\widetilde{\\alpha}_2 + \\widetilde{\\beta}$. There is no term involving $d\\log r \\wedge d\\theta$ because ${\\rm Res}_q(\\omega) = 0$. We have $\\alpha_1 := {\\rm Res}_r(\\omega) = \\iota_D^*(\\widetilde{\\alpha}_1)$ and $\\alpha_2 := {\\rm Res}_\\theta(\\omega) = \\iota_D^*(\\widetilde{\\alpha}_2)$. These are both closed because the residue maps ${\\rm Res}_r$ and ${\\rm Res}_\\theta$ are cochain morphisms. We have for $2 \\leq k \\leq n$ that\n\\begin{equation*}\n\t\\omega^k = (k d\\log r \\wedge \\widetilde{\\alpha}_1 \\wedge \\beta + k d\\theta \\wedge \\widetilde{\\alpha}_2 \\wedge \\beta + k(k-1) d \\log r \\wedge \\widetilde{\\alpha}_1 \\wedge d\\theta \\wedge \\widetilde{\\alpha}_2 + \\widetilde{\\beta}^2) \\wedge \\widetilde{\\beta}^{k-2}.\n\\end{equation*}\nNote here that $n \\geq 2$ is forced by nondegeneracy and the condition that ${\\rm Res}_q(\\omega) = 0$. Thus\n\\begin{equation*}\n\t{\\rm Res}_q(\\omega^k) = -k (k-1) \\alpha_1 \\wedge \\alpha_2 \\wedge \\beta^{k-2}, \\qquad 2 \\leq k \\leq n,\n\\end{equation*}\nwhere $\\beta := \\iota_D^*(\\widetilde{\\beta})$. In particular this shows that ${\\rm Res}_q(\\omega^2) = -{\\rm Res}_r(\\omega) \\wedge {\\rm Res}_\\theta(\\omega)$. Further, $\\rho$ is closed because ${\\rm Res}_q$ is a cochain morphism. Hence $\\rho$, $\\alpha_1$ and $\\alpha_2$ are invariantly defined forms on $D$, and $\\rho$ is a $2$-cosymplectic global pure spinor. We see that $\\rho$ is locally given by\n\\begin{equation*}\n\t\\rho = -\\alpha_1 \\wedge \\alpha_2 \\wedge e^\\beta.\n\\end{equation*}\nMoreover, we have that $\\rho_{2n-2} = -\\frac{1}{(n-2)!} \\alpha_1 \\wedge \\alpha_2 \\wedge \\beta^{n-2} \\neq 0$, due to nondegeneracy of $\\omega$.\n\\end{proof}\nThere is also a result for elliptic Poisson structures $\\pi$ whose elliptic residue is nonzero, in other words, for \\emph{nonzero elliptic Poisson structures} (for which $D$ is cooriented). Namely, in this case $D$ is a symplectic leaf of $\\pi$. See \\cite{KlaasseLanius18,KlaasseLi18} for more information.\n\nNote that in the presence of $n$ rational one-forms $\\alpha_i \\in \\Omega^1(M)$ whose cohomology classes are linearly independent, Tischler's theorem \\cite{Tischler70} implies that $M$ naturally fibers over $T^n$. An $n$-cosymplectic tuple whose one-forms satisfy this condition is called \\emph{proper}. Any $n$-cosymplectic tuple on a compact manifold can be perturbed slightly to be proper. Call a Poisson structure \\emph{proper} if its induced geometric structure on its degeneracy locus is proper. We obtain the following result for cooriented zero elliptic Poisson manifolds. This is the direct Poisson geometric analogue results in \\cite{BaileyCavalcantiGualtieri17} and a preliminary version of \\cite{CavalcantiGualtieri18} in the setting of generalized complex geometry.\n\\begin{prop} Let $(X,|D|,\\pi)$ be a cooriented zero elliptic Poisson manifold. Then $D$ carries a fibration over $T^2$, which can be made into a symplectic fibration after slight perturbation. Moreover, $\\pi$ can be perturbed through zero elliptic Poisson structures to be proper.\n\\end{prop}\n\\begin{rem} A similar result holds for log-Poisson manifolds (or $b^k$-Poisson) (\\cite{GuilleminMirandaPires14,MarcutOsornoTorres14,Cavalcanti17}): by \\autoref{prop:logcosymplectic} the degeneracy locus $Z$ carries a $1$-cosymplectic structure, hence fibers over $S^1$. This can be perturbed to be proper, making $Z$ a symplectic mapping torus. In \\cite{Cavalcanti17,MarcutOsornoTorres14} it is shown that $\\pi$ can be perturbed through log-Poisson structures so that $Z$ is proper. The argument used there proves essentially verbatim that the same holds for $b^k$-Poisson structure.\n\\end{rem}\n\\subsubsection{Relation with the Poisson residue}\nThere is a relation between the Lie algebroid residue maps that were used above, and the modular Poisson residues of \\autoref{sec:poissonmodulesmodfoliation} due to \\cite{GualtieriLi14,GualtieriPym13}.\n\nLet $\\mathcal{A} \\to X^{2n}$ be a Lie algebroid of smooth divisor-type and let $\\pi \\in {\\rm Poiss}(X)$ be of nondegenerate $\\mathcal{A}$-type with dual $\\mathcal{A}$-symplectic structure $\\omega$. This forces $I_\\pi = I_\\mathcal{A}$ by \\autoref{prop:apoissondivtypelifts}, and it ensures that $Z_\\mathcal{A} = Z_{I_\\mathcal{A}}$ is both the degeneraci locus of $\\mathcal{A}$, and the $(2n-2)$nd degeneracy locus of $\\pi$. Assuming that the restriction of $\\mathcal{A}$ to $Z_\\mathcal{A}$ is transitive with unimodular isotropy, and that $Z_\\mathcal{A}$ is of codimension-one, the residues ${\\rm Res}_{Z_\\mathcal{A}}(\\omega^n) \\in \\Omega^{2n-1}(Z_\\mathcal{A})$ and ${\\rm Res}_{n-1,{\\rm mod}}(\\pi) \\in \\mathfrak{X}^{2n-1}(Z_\\mathcal{A})$ are dual to each other in some sense. In particular, this happens for $b^k$-Poisson structures and their associated $\\mathcal{A}_Z^k$-symplectic structure. From \\autoref{prop:bkpoissoncosymplectic} we see that ${\\rm Res}_Z(\\omega^n) = \\alpha \\wedge \\beta^{n-1}$ if locally $\\omega = dz\/z^k \\wedge \\widetilde{\\alpha} + \\widetilde{\\beta}$ for $z \\in j_{k-1}$. On the other hand, we have that ${\\rm Res}_{n-1,{\\rm mod}}(\\pi) = v \\wedge \\pi_Z^{n-1}$ for $v$ a local modular vector field. Note that $\\beta$ is not globally defined, and nor is $v$. However, their pure spinors make sense, and define the same Dirac structure (the forms we used above as spinors are also called contravariant spinors, with multivector fields being covariant spinors, see \\cite{AlekseevBursztynMeinrenken09,Klaasse18}). See also the discussions in \\cite[Proposition 1.8]{GualtieriLi14} and \\cite[Remark 7.3]{GualtieriPym13}. In general the relation between the Poisson modular residue and the residue maps of the associated symplectic Lie algebroid can be more involved. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nIn the Minimal Supersymmetric Standard Model (MSSM)~\\cite{R1}, if the mixing\nbetween third generation squarks is large, stops\/sbottoms can be rather\nlight~\\cite{qmix} and at the same time, their coupling to Higgs bosons can \nbecome substantial.  This might have a rather large impact on the phenomenology\nof the MSSM Higgs bosons~$^{\\rm 3-7}$. More precisely, concentrating on \nscalar top quarks, their mass eigenvalues are given by\n\\begin{equation}\nm_{\\tilde{t}_{1,2}}^2 = m_t^2 + \\frac{1}{2} \\left[ m_{\\tilde Q_L}^2 + m_{\\tilde\nt_R}^2 +\\cdots \\mp \\sqrt{ (m_{\\tilde Q_L}^2 - m_{\\tilde t_R}^2 +\\cdots)^2 + \n4m_t^2 \\tilde{A}_t^2 } \\right]\\;, \n\\end{equation}\nwhere $m_{\\tilde Q_L}$, $m_{\\tilde t_R}$ are the soft-SUSY breaking scalar \nmasses and the dots stand for the $D$--terms $\\propto M^2_Z \\cos 2\\beta$.  In the \ndecoupling limit (the lightest $h$ boson is Standard Model like and the other \nbosons $A,H$ and $H^\\pm$, are very heavy), the expressions of the coupling $h\n\\tilde{t_1} \\tilde{t_1}$ simply reads ($\\theta_t$ is the mixing angle and $s_W\n\\equiv \\sin\\theta_W$)\n\\begin{equation}\ng_{h \\tilde{t}_1 \\tilde{t}_1 } = \\cos 2\\beta \\left[ \\frac{1}{2} \\cos^2 \\theta_t\n- \\frac{2}{3} s^2_W \\cos 2 \\theta_t \\right] + \\frac{m_t^2}{M_Z^2} + \\frac{1}{2}\n\\sin 2\\theta_t \\frac{m_t \\tilde{A}_t } {M_Z^2}\\;.  \\label{ghtt}\n\\end{equation}\nLarge values of $\\tilde{A}_t\\equiv A_t -\\mu\/ \\tan\\beta$ lead to \n$g_{h \\tilde{t}_1 \\tilde{t}_1} \\sim \\tilde{A}_t$  \nand to an almost maximal $\\tilde{t}$ \nmixing angle, $|\\sin 2 \\theta_t| \\simeq 1$, in particular\nif $m_{\\tilde Q_L} \\simeq m_{\\tilde t_R}$.  \nThe measurement of this important coupling would open a\nwindow to probe directly some of the soft--SUSY breaking terms of the\npotential. To measure Higgs--squarks couplings directly, one needs to consider\nthe three--body associated production of Higgs bosons with scalar quark\npairs~$^{\\rm 4-7}$. This is the supersymmetric analog to the processes\nof Higgs boson radiation from top quark lines~\\cite{ppttH,eetth} which allows \nto probe the $t\\bar{t}$--Higgs Yukawa coupling directly. [At the LHC, the \nprocess $pp \\to t\\bar{t}+$Higgs~\\cite{ppttH} although not competitive with the \ngluon fusion mechanism~\\cite{R7}, can provide a complementary signal since \nbackgrounds are smaller.] \n\nHere, we report on the production of a light Higgs boson $h$ in association\nwith top squarks at future $e^+e^-$ linear machines~\\cite{Xtth}, including the\n$\\gamma \\gamma$ option, both in the unconstrained MSSM and minimal SUGRA cases.\nFor simplicity, we work in the approximation of being close to the decoupling \nlimit, which  implies that we do not consider other Higgs production processes \nwith e.g. the heavy $H$ or $A$ bosons produced in association \nwith top squarks. For large masses, these processes will be suppressed by \nphase--space well before the decoupling regime is reached.\n\\section{Associated production at e$^+$e$^-$ colliders} \nAt future linear $e^+e^-$ colliders, the final state $\\tilde{t}_1 \\tilde{t}_1 h$\nmay be generated in three ways: $(i)$ two--body production of a mixed pair\nof top squarks and the decay of the heaviest stop to the lightest one and a \nHiggs boson, $(ii)$ the continuum production in $e^+e^-$ annihilation $e^+e^- \\rightarrow\n\\tilde{t}_1 \\tilde{t}_1h$ and $(iii)$  the continuum production in \n$\\gamma \\gamma$ collisions $\\gamma \\gamma \\rightarrow \\tilde{t}_1 \\tilde{t}_1h$.\n\n\\vspace*{-4mm}\n\n\\subsection{Two--body production and decay} \nThe total cross section~\\cite{R8} for the process $(i)$ should, in principle, \nbe large enough for the final state to be copiously produced. However, $\\sigma( \ne^+e^- \\rightarrow \\tilde{t}_1 \\tilde{t}_2$) involves the $Z\\tilde{t}_1 \\tilde{t}_2$ \ncoupling, proportional to $\\sin 2 \\theta_t$, while the $\\vert g_{h \\tilde{t}_1 \n\\tilde{t}_2}\\vert $ coupling  in most of the parameter space is proportional \nto $\\cos 2\\theta_t$, such that the cross section times branching ratio will be \nvery small in the no mixing [$\\theta_t \\sim 0$] and maximal mixing $[|\\theta_t| \n\\sim \\pi\/4]$ cases. \n[In addition, the decay width $\\tilde{t}_2 \\rightarrow h\\tilde{t}_1$ is \nin general much smaller~\\cite{R8} than the $\\tilde{t}_2$ decay widths into \nchargino and neutralinos]. Nevertheless, there are regions of the MSSM \nparameter space where\nthe combination $\\sin 2 \\theta_t \\times \\cos 2 \\theta_t$ can be large; this\noccurs typically for a not too small $m_{\\tilde t_L}$--$m_{\\tilde t_R}$ \nsplitting and moderate $\\tilde A_t$ values.\\cite{Xtth} In this case, which is \noften realized in the mSUGRA scenario, this mechanism is visible for the \nexpected high luminosities\\cite{TESLA} $\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$.  \n\nThis is illustrated in Fig.~1, where the cross section $e^+e^- \\rightarrow \\tilde{t}_1 \n\\tilde{t}_2$ times the branching ratio BR($\\tilde{t}_2 \\rightarrow \\tilde{t}_1 h)$ is \nshown as a function of the $\\tilde{t}_1$ mass at a c.m. energy of $\\sqrt{s}\n=800$ GeV. We have chosen a mSUGRA scenario with $\\tan\\beta=30$, $m_{1\/2} =$ \n100 GeV, $A_0 = -600$ GeV and sign$(\\mu)= +$. (The dotted lines show the \ncontribution of the  non--resonant process, discussed below, for the same \ninput choice). The cross section can reach the level of 1 fb for relatively\nsmall $m_{\\tilde{t}_1}$ values, leading to thousand events in a few years, for \n$\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$. \n\\begin{figure}[htb]\n\\vspace{-.5cm}\n\\begin{center}\n\\mbox{\n\\psfig{figure=ee_sug_1.eps,width=10cm}}\n\\end{center}\n\\vspace{-9.cm}\n\\caption[]{The production cross section $\\sigma (e^+e^- \\rightarrow \\tilde{t}_1 \n\\tilde{t}_1 h$) [in fb] as a function of $m_{\\tilde{t}_1}$ in the mSUGRA \ncase; $\\tan\\beta=$ 30, $m_{1\/2} =$ 100 GeV and $A_0 =-600$ GeV.}\n\\end{figure}\n\\subsection{Production in the continuum in e$^+$e$^-$ collisions}\nThe cross section for the process $e^+e^- \\rightarrow \\tilde{q}_i \\tilde{q}_i \\Phi$ with \n$\\Phi$ the CP--even Higgs boson $h$ or $H$, and $\\tilde{q}_i$ any of the two \nsquarks has been calculated in \\cite{Xtth}.\nWe show in Fig.~2 the rates for the $\\tilde{t}_1 \\tilde{t}_1 h$ final state  as\na function of the $\\tilde{t}_1$ mass in the unconstrained MSSM, at\n$\\sqrt{s}=800$ GeV.  For not too large $\\tilde{t}_1$ masses and large values of\nthe parameter $\\tilde A_t$, the production cross sections can exceed  \n$1$ fb, to be compared to the\nSM--like process $e^+e^- \\rightarrow t \\bar{t}h$ \\cite{eetth} of the order of 2 fb for\n$M_h \\sim 130$ GeV.  This provides more than one thousand events in a few\nyears, with a luminosity  $\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$, which should\nbe sufficient to isolate the final state and measure $g_{\\tilde{t}_1\n\\tilde{t}_1 h}$  with some accuracy.  \n\\begin{figure}[htbp]\n\\vspace{-5mm}\n\\begin{center}\n\\mbox{\n\\psfig{figure=ee_mt1.eps,width=10cm}}\n\\end{center}\n\\vspace*{-9cm}\n\\caption[]{The cross section $\\sigma (e^+e^- \\rightarrow \\tilde{t}_1 \\tilde{t}_1 h$) \n[in fb] as a function of the $\\tilde{t}_1$ mass and the choices $\\tan\\beta=30$, \n$A_t=$ 0 (0.5) TeV; $\\tan\\beta=3$, $A_t=$ 1.2 TeV (and $\\mu =-$600 GeV).} \n\\end{figure}\n\nNote however that $\\tilde A_t$ cannot be arbitrarily large without conflicting\npresent constraints: more precisely, the absence of charge and color breaking\nminima (CCB) \\cite{CCB} can put rather stringent bounds on $\\tilde{A}_t$, and\nlarge $\\tilde{A}_t$ values also generate potentially large contributions to\nelectroweak high--precision observables, in particular to the $\\rho$ parameter\n\\cite{drho}, severely constrained  by LEP1 data.\\cite{LEPrho} \nThose constraints, as well as the present \nexperimental lower bounds on the top squark and Higgs \nboson masses, were systematically taken into account in our analysis.\nWe observed\nthat in the unconstrained MSSM case, the continuum production cross section in\n$e^+e^-$ annihilation $\\sigma(e^+e^- \\rightarrow \\tilde{t}_1 \\tilde{t}_1 h$) is often larger\nthan the resonant cross section for the production of $\\tilde{t}_1 \\tilde{t}_2$\nand the subsequent 2--body decay $\\tilde{t}_2 \\rightarrow \\tilde{t}_1 h$, but this is\nnot generic. Indeed, in a situation where both a non-negligible $m_{\\tilde\nt_L}$--$m_{\\tilde t_R}$ splitting and a moderate $\\tilde A_t$ occurs, provided\nthere is sufficient phase space allowed, the production via a resonant $\\tilde\nt_2$ becomes competitive and even dominant, as illustrated in Fig. 1.\n\n\\vspace*{-4mm}\n\n\\subsection{Production in the continuum in $\\gamma \\gamma$ collisions} \nFuture high--energy $e^+e^-$ linear colliders can be turned into high--energy \n$\\gamma \\gamma$ colliders, with the high energy photons coming from Compton\nback--scattering of laser beams.\\cite{laser} The c.m. energy of the $\\gamma \n\\gamma$ collider is expected to be as much as $\\sim 80\\%$ of the one of the \noriginal $e^+e^-$ machine. However, the total luminosity is expected to be \nsomewhat smaller than the one of the $e^+e^-$ mode.\n\\begin{figure}[htb]\n\\begin{center}\n\\vspace{-.5cm}\n\\mbox{\n\\psfig{figure=gaga_mt1.eps,width=10cm}}\n\\end{center}\n\\vspace{-9cm}\n\\caption[]{$\\sigma (\\gamma\\gamma \\rightarrow \\tilde{t}_1 \n\\tilde{t}_1 h$) [in fb] at $\\sqrt{s_{\\gamma \\gamma}} =600$ GeV as a function \nof $m_{\\tilde{t}_1}$. The other parameters have the same values as in Fig.~2.}\n\\end{figure}\nThe total cross section for the subprocess $\\gamma \\gamma \\rightarrow \\tilde{t}_1\n\\tilde{t}_1 h$, calculated in \\cite{Xtth}, is shown in Fig.~3 at a \ntwo--photon c.m. energy $\\sqrt{s}_{\\gamma \\gamma}  \n\\raisebox{-0.13cm}{~\\shortstack{$<$ \\\\[-0.07cm] $\\sim$}}~ 0.8 \\sqrt{s}_{ee} =600$ GeV and as a function of the\n$\\tilde{t}_1$ mass, without convolution with the photon spectrum and with the\nsame inputs and assumptions as in Fig.~2 to compare with the $e^+e^-$ mode. \nBecause the c.m. energy of the $\\gamma \\gamma$ collider is only $\\sim 80\\%$ of\nthe one of the original $e^+e^-$ machine, the process is of course less\nphase--space favored than in the $e^+e^-$ mode.  Nevertheless, the cross section\nfor the $\\tilde{t}_1 \\tilde{t}_1h$ final state is of the same order as in the\n$e^+e^-$ mode for c.m. energies not too close to the kinematical threshold, and\nthe process might be useful to obtain complementary information since it does\nnot involve the $Z$--boson and $\\tilde{t}_2$ exchanges.  If the luminosities of\nthe $\\gamma \\gamma$ and $e^+e^-$ colliders are comparable, a large number of\nevents might be collected for small stop masses and large \n$\\tilde A_t$ values.  \n\n\\vspace*{-4mm}\n\n\\subsection{Decay modes and signal}\nTop squarks in the mass range discussed above will mainly decay~\\cite{R8} into \na $c$ quark + neutralino, $\\tilde{t}_1 \\to c\\chi_1^0$, or a $b$ quark + \nchargino,\n$\\tilde{t}_1 \\rightarrow b\\chi^+$.  In this latter case the lightest chargino,\n$\\chi_1^+$  will decay into the LSP and a real or virtual $W$ boson, leading to\nthe same topology as in the case of the top quark decay, but with a large\namount of missing energy due to the undetected LSP\n(three and four--body decays~\\cite{fourbody} of the stop are also\npossible with the same topology). At $e^+e^-$ colliders one can\nuse the dominant decay mode of the lightest Higgs boson, $h \\rightarrow b\\bar{b}$. The\nfinal state topology will then consist of $4b$ quarks, two of them peaking at\nan invariant mass $M_h$, two real (or virtual) $W$'s and missing energy. With\nefficient micro--vertex detectors, this final state should be rather easy to \ndetect.  \n\n\\vspace*{-4mm}\n\n\\section{Conclusions} \nAt $e^+e^-$ colliders with c.m. energies $\\sqrt{s} \\raisebox{-0.13cm}{~\\shortstack{$>$ \\\\[-0.07cm] $\\sim$}}~ 500$ GeV and with very\nhigh luminosities $\\int {\\cal L}{\\rm d}t \\sim 500$ fb$^{-1}$, the process $e^+e^- \\rightarrow\n\\tilde{t}_1 \\tilde{t}_1h$ can lead to several hundreds of events, since the\ncross sections can exceed the level of a 1 fb for not too heavy top squarks and\nlarge trilinear coupling, $\\tilde A_t \\raisebox{-0.13cm}{~\\shortstack{$>$ \\\\[-0.07cm] $\\sim$}}~ 1$ TeV.  In the case where the top\nsquark decays into a $b$ quark and a real\/virtual chargino, the final state\ntopology with $4b$ quarks, missing energy and additional jets or leptons will\nbe rather spectacular and should be easy to be seen experimentally, thanks to\nthe clean environment of these colliders.  In the $\\gamma \\gamma$ option of the\n$e^+e^-$ collider, the cross sections are similar as previously far from the\nparticle thresholds, but are suppressed for larger masses because of the\nreduced c.m. energies; for $\\gamma \\gamma$ luminosities of the same order as\nthe original $e^+e^-$ luminosities, the $\\tilde{t}_1 \\tilde{t}_1h$ final state\nshould also be observable in this mode, at least in some areas of the MSSM \nparameter space.  \n\nThe production cross section of the $\\tilde{t}_1 \\tilde{t}_1 h$ final state \nis directly proportional to the square of the $\\tilde{t}_1 \\tilde{t}_1 h$\ncouplings, therefore studying this process will allow to measure this important\ncoupling and to probe directly some of the soft--SUSY breaking parameters.  \n\n\\vspace*{-4mm}\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}