diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmcnd" "b/data_all_eng_slimpj/shuffled/split2/finalzzmcnd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmcnd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\nThe classical definition of martingales is extended to a more general\ncase in the space of Banach lattices by V.~Troitsky \\cite\n{TroitskyMartingales:05}. In the Banach lattice framework,\nmartingales are defined without a probability space and the famous\nDoob's convergence theorem was reproduced. Moreover, under certain\nconditions on the Banach lattice, it was shown that the set of bounded\nmartingales forms a Banach lattice with respect to the point-wise order.\nIn 2011, H.~Gessesse and V.~Troitsky \\cite{GessesseMartingales:11}\nproduced several\\vadjust{\\goodbreak} sufficient conditions for the space of bounded\nmartingales on a Banach lattice to be a Banach lattice itself. They\nalso provided examples showing that the space of bounded martingales is\nnot necessarily a vector lattice. Several other works have been done by\nother authors with regard to martingales in vector lattices, such as\n\\cite{Watson:13, Grobler:15}.\n\nIn the theory of random processes,\nnot just the study of martingale convergence is important,\nbut the study of convergence of martingale-like stochastic sequences and processes, and the determination of interrelation between them are also crucial.\nSo it is natural to ask if martingale-like\nsequences can be defined in a vector lattice or Banach lattice\nframework. In this article, we define and study martingale-like\nsequences in Banach lattices along the same lines as martingales are\ndefined and studied in \\cite{TroitskyMartingales:05}.\n\nClassically, a martingale-like sequence is defined as follows (for\ninstance, see a paper by A.~Melnikov \\cite{Melnikov:82}). Consider a\nprobability space $(\\varOmega,\\mathcal{F},P)$ and a filtration\n$(\\mathcal{F}_n)_{n=1}^{\\infty}$, i.e., an increasing sequence of\ncomplete sub-sigma-algebras of $\\mathcal{F}$. An integrable stochastic\nsequence $x=(x_n,\\mathcal{F}_n)$ is an {\\bf$L^1$-martingale} if\n\\[\n\\lim_{n\\rightarrow\\infty} \\sup_{m\\geq n}E\\big|E(x_m |\n\\mathcal {F}_n) - x_n\\big| = 0.\n\\]\nAn integrable stochastic sequence $x=(x_n,\\mathcal{F}_n)$ is an {\\bf\n$E$-martingale} if\n\\[\nP\\bigl\\{ \\omega: E(x_{n+1} | \\mathcal{F}_n) \\neq\nx_n \\text{ infinitely often } \\bigr\\}=0.\n\\]\n\nHere we extend the definition of $L^1$-martingales and $E$-martingales\nin a general Banach lattice $X$ following the same lines as the\ndefinition of martingales in Banach lattices in \\cite\n{TroitskyMartingales:05}. First we mention some terminology and\ndefinitions from the theory of Banach lattices for the reader\nconvenience. For more detailed exploration, we refer the reader to\n\\cite{Aliprantis:85}. A {\\bf vector lattice} is a vector space\nequipped with a lattice order relation, which is compatible with the\nlinear structure. A {\\bf Banach lattice} is a vector lattice with a\nBanach norm which is monotone, i.e., $0\\leq x \\leq y$ implies $\\norm\n{x}\\leq\\norm{y}$, and satisfies $\\norm{x}=\\querymark{Q1} \\lVert\\abs{x}\n\\rVert$ for any two vectors $x$ and $y$. A vector lattice is said to\nbe {\\bf order complete} if every nonempty subset that is bounded above\nhas a supremum. We say that a Banach lattice has {\\bf order continuous\nnorm} if $\\norm{x_{\\alpha}}\\rightarrow0$ for every decreasing net\n$(x_{\\alpha})$ with $\\inf x_{\\alpha}=0$. A Banach lattice with order\ncontinuous norm is order complete. A sublattice $Y$ of a vector lattice\nis called an (order) {\\bf ideal} if $y\\in Y$ and $|x|\\leq|y|$ imply\n$x\\in Y$. An ideal $Y$ is called a {\\bf band} if\n$x = \\sup_{ \\alpha} x_{\\alpha}$ implies $x\\in Y$ for every positive\nincreasing net $(x_{\\alpha})$ in $Y$. Two elements $x$ and $y$ in a\nvector lattice are said to be {\\bf disjoint} whenever $|x|\\wedge|y| =\n0$ holds. If $J$ is a nonempty subset of a vector lattice, then its\n{\\bf disjoint complement} $J^d$ is the set of all elements of the\nlattice, disjoint to every element of $J$. A band $Y$ in a vector\nlattice $X$ that satisfies $X = Y\\otimes Y^d$ is refered to as a {\\bf\nprojection band}. Every band in an order complete vector lattice is a\nprojection band. An operator $T$ on a vector lattice X is positive if\n$Tx\\geq0$ for every $x\\geq0$. A sequence of positive projections\n$(E_n)$ on a vector lattice $X$ is called a {\\bf filtration} if $E_nE_m\n= E_{n\\wedge m}$. A sequence of positive contractive projections\n$(E_n)$ on a normed lattice $X$ is called a {\\bf contractive\nfiltration} if $E_nE_m = E_{n\\wedge m}$. A~filtration $(E_n)$ in a\nnormed lattice $X$ is called \\term{dense} if $E_nx\\rightarrow x$ for\neach $x$ in $X$. In many articles such as in \\cite\n{TroitskyMartingales:05}, a \\term{martingale} with respect to a\nfiltration $(E_n)$ in a vector lattice $X$ is defined as a sequence\n$(x_n)$ in $X$ such that $E_nx_m=x_n$ whenever $m\\ge n$.\n\n\\section{Main definitions}\n\n\\begin{definition}\\label{def1}\nA sequence $ (x_n)$ of elements of a normed lattice $X$ is called an\n{\\bf $X$-martingale} relative to a contractive filtration $(E_n)$ if\n\\[\n\\lim\\limits\n_{n\\rightarrow\\infty} \\sup_{m\\geq n}\\norm{E_nx_m\n- x_n} = 0.\n\\]\n\\end{definition}\n\n\\begin{definition}\\label{def2}\nA sequence $ (x_n)$ of elements of a vector lattice $X$ is called an\n$\\mathcal{E}$-\\textbf{martingale} relative to a filtration $(E_n)$ if\nthere exists $n\\geq1$ such that $E_m x_{m+1}=x_m$ for all $m\\geq n.$\n\\end{definition}\n\nNote that Definition~\\ref{def2} is equivalent to saying a sequence\n$(x_n)$ is an $\\mathcal{E}$-martingale if there exists $l\\geq1$ such\nthat $E_n x_{m}=x_n$ whenever $m\\geq n \\ge l$. The symbol ``$\\mathcal\n{E}$'' stands for eventual so when we say $(x_n)$ is an $\\mathcal\n{E}$-martingale, we are saying that after a first few finite elements\nof the sequence, the sequence becomes a martingale.\n\nSequences defined by Definition~\\ref{def1} and Definition~\\ref{def2}\nare collectively called {\\bf martin\\-gale-like sequences}.\nNotice that every martingale $(x_n)$ in a vector lattice $X$ with\nrespect to a filtration $(E_n)$ is obviously an $\\mathcal\n{E}$-martingale with respect to the filtration\n$(E_n)$. Moreover, every $\\mathcal{E}$-martingale $(x_n)$ in a Banach\nlattice $X$ with respect to a contractive filtration $(E_n)$ is an\n$X$-martingale with respect to the contrative filtration\n$(E_n)$.\nNote that for every $x$ in a vector lattice $X$ and a filtration\n$(E_n)$ in $X$, the sequence $(E_n x)$ is an $\\mathcal{E}$-martingale\nwith respect to the filtration $(E_n)$. If $x$ is in a normed space $X$\nand $(E_n)$ is a contractive filtration, then the sequence $(E_n x)$ is\nan $X$-martingale with respect to the contractive filtration $(E_n)$.\n\nBy considering any nonzero martingale $(x_n)$ in a Banach lattice $X$\nwith respect to filtration $(E_n)$ where $x_1$ is nonzero without loss\nof generality, we can define a sequence $(y_n)$ such that $y_1=2x_1$\nand $y_n=x_{n}$ for all $n\\geq2$. Then one can see that $(y_n)$ is an\n$\\mathcal{E}$-martingale with respect to the filtration $(E_n)$.\nHowever, $(y_n)$ is not a martingale.\n\nNote that every sequence which converges to zero is an $X$-martingale\nwith respect to any contractive filtration $(E_n)$ because if\n$x_n\\rightarrow0$ and $m> n$ then\n$\\norm{E_n x_m - x_n}\\leq\\norm{x_m} +\\norm{x_n} \\rightarrow0$ as\n$n\\rightarrow\\infty$. So one can easily create an $X$-martingale\n$(x_n)$ which is not $\\mathcal{E}$-martingale by setting $x_n=\\frac\n{1}{n}x$ where $x$ is a nonzero vector in $X$.\n\nA martingale-like sequence $A=(x_n)$ with respect to a contractive\nfiltration $(E_n)$ on a normed lattice $X$ is said to be {\\bf bounded}\nif its norm defined by $\\norm{A}=\\sup_n\\norm{x_n}$ is finite. Given\na contractive filtration $(E_n)$ on a normed lattice $X$, we denote the\nset of all bounded $X$-martingales with respect to the contractive\nfiltration $(E_n)$ by $M_X=M_X(X,(E_n))$ and the set of all bounded\n$\\mathcal{E}$-martingales with respect to the contractive filtration\n$(E_n)$ by $M_E=M_E(X,(E_n))$. With the introduction of the sup norm in\nthese spaces, one can show that $M_X$ and $M_E$ are normed spaces.\nKeeping the notation $M$ of \\cite{TroitskyMartingales:05} for all\nbounded martingales with respect to the contractive filtration $(E_n)$\nand from the preceding arguments, these spaces form a nested increasing\nsequence of linear subspaces $M \\subset M_E \\subset M_X \\subset\\ell\n_\\infty(X)$, with the norm being exactly the $\\ell_\\infty(X)$ norm.\\looseness=-1\n\n\\begin{theorem}\\label{Mx-BS}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$, then\nthe collection of $X$-martingales $M_X$ is a closed subspace of $\\ell\n_\\infty(X)$, hence a Banach space.\\looseness=-1\n\\end{theorem}\n\\begin{proof}\nSuppose a sequence $(A^m)=(x^m_n)$ of $X$-martingales converges to $A$\nin $\\ell_\\infty(X)$. We show $A$ is also an $X$-martingale. Indeed,\nfrom $\\norm{A^m-A}=\\sup_n\\norm{x^m_n-x_n}\\rightarrow0$ as\n$m\\rightarrow\\infty$, we have that for each $n\\geq1$, $\\norm\n{x^m_n-x_n}\\rightarrow0$ as $m\\rightarrow\\infty$.\nNote that for $l\\ge n$,\n\\begin{align*}\n\\norm{E_nx_l-x_n}&=\\norm{E_nx_l-E_nx^m_l+E_nx^m_l-x^m_n+x^m_n-x_n}\n\\\\\n&\\leq\\norm{E_nx_l-E_nx^m_l}\n+\\norm{E_nx^m_l-x^m_n}\n+\\norm{x^m_n-x_n}.\n\\end{align*}\nFrom these inequalities and the contractive property of the filtration,\nwe have\n\\[\n\\lim_{n\\rightarrow\\infty} \\sup_{l\\geq n}\\norm{E_nx_l\n- x_n} = 0.\\qedhere\n\\]\n\\end{proof}\n\n\\begin{corollary}\\label{inclusion}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$, then\n$\\overline{M_E} \\subset M_X.$\n\\end{corollary}\n\n\\begin{lemma}\\label{conv-mx}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$ and\n$A=(x_n)$ be in $M_X$ where $x_n\\rightarrow x$. Then\n\\[\n\\lim\\limits\n_{n\\rightarrow\\infty} \\sup_{m\\ge n}\\norm{E_mx -\nx_m} = 0.\n\\]\n\\end{lemma}\n\n\\begin{proof}\nLet $A=(x_n)$ be in $M_X$ where $x_n\\rightarrow x$. Thus, for $m\\geq n$\n\\[\n\\norm{E_nx - x_n}= \\norm{E_nx -\nE_nx_m+E_nx_m-x_n}\n\\leq\\norm{x-x_m} +\\norm{E_nx_m-x_n}.\n\\]\nTaking $\\lim\\limits_{n\\rightarrow\\infty} \\sup_{m\\geq n}$ on both\nsides of the inequality completes the proof.\n\\end{proof}\n\nThe following proposition confirms that for any convergent element\\querymark{Q2}\n$A=(x_n)$ of $M_X$ we can find a sequence in $M_E$ that converges to $A$.\n\\begin{proposition}\\label{halfdense}\nLet $(E_n)$ be a contractive filtration on a Banach lattice $X$ and\n$A=(x_n)$ be a sequence in $M_X$ such that $x_n\\rightarrow x$. Then\nthere exists a sequence $A^m$ in $M_E$ such that $A^m \\rightarrow A$ in\n$\\ell_{\\infty}(X)$.\n\\end{proposition}\n\\begin{proof}\nSuppose $x_n \\rightarrow x$ as $n\\rightarrow\\infty$. First note that\nthe sequence $(E_nx)$ is in $M$. Now define $A^m=(a^m_n)$ such that\n\\[\na^m_n=\n\\begin{cases}\n x_n,& \\text{for } n\\le m,\\\\\n E_nx,& \\text{for } n > m.\n\\end{cases}\n\\]\nThen $A^m\\in M_E$ and $A^m \\rightarrow A$ in $\\ell_{\\infty}(X)$,\nhence $A\\in\\overline{M_E}$. Indeed, by Lemma~\\ref{conv-mx},\n\\[\n\\lim_{m\\rightarrow\\infty}\\norm{A^m-A}=\\lim_{m\\rightarrow\\infty\n}\n\\sup_j\\norm{E_{m+j}x-x_{m+j}}=0.\\qedhere\n\\]\n\\end{proof}\n\nIn \\cite{TroitskyMartingales:05} and \\cite{GessesseMartingales:11}\nseveral sufficient conditions are es\\querymark{Q3}tablished where the set of bounded\nmartingales $M$ is a Banach lattice. In \\cite{GessesseMartingales:11},\ncounter examples are provided where $M$ is not a Banach lattice. So,\none may similarly ask when are $M_X$ and $M_E$ Banach spaces and Banach\nlattices? We start by showing a counter example that illustrates that\n$M_E$ is not necessarily a Banach space.\n\\begin{example}\nLet $c_0$ be the set of sequences converging to zero. Consider the\nfiltration $(E_n)$ where $E_n \\sum_{i=1}^{\\infty} \\alpha_i e_i =\n\\sum_{i=1}^{n} \\alpha_i e_i$. Thus the sequence $(y_n)$ where $y_n=\n\\sum_{i=1}^{n} \\frac{1}{i} e_i$ is an $E$-martingale with respect to\nthis filtration. We define a sequence of $E$-martingales $A^m$ as\n$A^m=(x_n^m)$ where\n\\[\nx_n^m=\n\\begin{cases}\n \\sum_{i=n}^{\\infty} \\frac{1}{i} e_i ,& \\text{for } n\\leq m,\\\\\n y_n\/m,& \\text{for } n>m.\n\\end{cases}\n\\]\nDefine a sequence $A=(x_n)$ where $x_n=\\sum_{i=n}^{\\infty} \\frac\n{1}{i} e_i $. We can see that $A$ is not an $E$-martingale. But one can\nshow that $A^m$ converges to $A$. Indeed,\n\\[\n\\bigl\\lVert A^m - A \\bigr\\rVert=\\sup_{n}\n\\big\\|x^m_n-x\\big\\|=\\sup_{n \\in\n\\{m+1,m+2, \\ldots\\}}\n\\Bigg\\|y_n\/m-\\sum_{i=n}^{\\infty}\n\\frac{1}{i} e_i \\Bigg\\| \\rightarrow0\n\\]\nas $m\\rightarrow\\infty.$\n\\end{example}\n\n\\section{When is $M_E$ a vector lattice?}\n\nGiven a vector (Banach) lattice $X$ and a filtration (respectively\ncontractive) $(E_n)$ on $X$, we can introduce order structure on the\nspaces $M_E$ and $M_X$ as follows. For two bounded $\\mathcal\n{E}$-martingales (respectively $X$-martingales) $A=(x_n)$ and\n$B=(y_n)$, we write $A\\geq B$ if $x_n\\geq y_n$ for each $n$. With this\norder $M_E$ and $M_X$ are ordered vector spaces and the monotonicity of\nthe norm follows from the monotonicity of the norm of $X$, i.e. for two\n$\\mathcal{E}$-martingales (respectively $X$-martingales) with $0\\leq A\n\\leq B$, we have $\\norm{A}\\leq\\norm{B}$. For two $\\mathcal\n{E}$-martingales (respectively $X$-martingales) $A=(x_n)$ and\n$B=(y_n)$, one may guess that $A\\lor B$ (or $A\\wedge B$) can be\ncomputed by the formulas $A\\lor B =(x_n\\lor y_n)$ (or $A\\wedge\nB=(x_n\\wedge y_n)$). We show in the following theorem that this is in\nfact the case in order for $M_E$ to be a vector lattice. However, this\nis not obvious to show in the case of $M_X$.\n\n\\begin{theorem}\\label{vl-equivalence}\nLet $X$ be a vector lattice. Then the following statements are equivalent.\n\\begin{enumerate}\n\\item[(i)]$M_E$ is a vector lattice.\n\\item[(ii)] For each $A=(x_n)$ in $M_E$, the sequence $(|x_n|)$ is an\n$\\mathcal{E}$-martingale and $\\abs{A}=(\\abs{x_n})$.\n\\item[(iii)] $M_E$ is a sublattice of $\\ell_{\\infty}(X)$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nFirst we show (\\textit{i})~$\\implies$~(\\textit{ii}). Suppose $M_E$ is a vector lattice and\n$A=(x_n)$ is in $M_E$. Since $M_E$ is a vector lattice, $\\abs{A}$\nexists in $M_E$, say $|A|=B:=(y_n)$. Since $\\pm A\\le B$, for each $n$,\n$\\pm x_n\\le y_n$. So, $\\abs{x_n}\\le y_n$ for each $n$. Since $B$ is in\n$M_E$, there exists $l$ such that $E_ny_m=y_n$ whenever $m\\ge n\\ge l$.\nNow we claim that $y_n=|x_n|$ for each $n$. Fix $k> l$. We show\n$y_n=|x_n|$ for each $n\\le k$.\n\nIndeed, define an $\\mathcal{E}$-martingale $C=(z_n)$ where\n\\[\nz_n=\n\\begin{cases}\n \\abs{x_n},& \\text{for } n\\le k,\\\\\n y_n, &\\text{for } n> k.\n\\end{cases}\n\\]\nSince $k>l$ we can easily see that $C$ is an $\\mathcal{E}$-martingale.\nMoreover, $C\\ge0$ and $\\pm A \\le C \\le B$. Since $\\abs{A}=B$, $C=B$.\nThus, for every $n\\le k$, $y_n=|x_n|$. This establishes (\\textit{ii}).\n\n(\\textit{ii})~$\\implies$~(\\textit{iii})~$\\implies$~(\\textit{i}) is straightforward.\n\\end{proof}\n\nUsing the equivalence in Theorem~\\ref{vl-equivalence}, the following\nexamples illustrate\nthat $M_E$ is not always a vector lattice.\n\n\\begin{example}\\label{ReviewerExample}\nConsider the classical martingale $(x_n)$ in $L_1[0,1]$ where\n$x_n=\\break2^n\\mathbf{1}_{[0,2^{-n}]} -\\mathbf{1}$ with the filtration\n$(\\mathcal{F}_n)$ where $\\mathcal{F}_n$ is the smallest sigma algebra\ngenerated by the set\n\\[\n\\bigl\\{ \\bigl[0,2^{-n}\\bigr], (2^{-n}, 2^{-n+1}],\n\\dots, (1-2^{-n}, 1] \\bigr\\}.\n\\]\nOne can easily show that\n\\[\nE_n\\abs{x_{n+1}}=E \\bigl[|x_{n+1}| |x_n \\bigr]\n\\ne|x_n|\n\\]\nfor\nevery $n$ and the sequence $(|x_n|)$ fails to be an $\\mathcal\n{E}$-martingale. Hence, Theorem~\\ref{vl-equivalence} implies that\n$M_E$ is not a vector lattice.\n\\end{example}\n\n\\begin{example}\\label{cexample}\nConsider the filtration $(E_n)$ defined on $c_0$ as follows. For each\n$n=0, 1, 2, \\ldots$\n\\[\nE_n =\n\\begin{bmatrix}\n1 & & & & & & & \\\\\n& \\ddots & & & & & &\\\\\n& & 1 & & & & &\\\\\n& & &1\/2 &1\/2 & & &\\\\\n& & &1\/2 &1\/2 & & &\\\\\n& & & & &1\/2 &1\/2 &\\\\\n& & & & &1\/2 &1\/2 &\\\\\n& & & & & & &\\ddots\\\\\n\\end{bmatrix}\n\\]\nwith $2n$ ones in the upper left corner. For each $e_i=(0,\\ldots, 0,\n\\underbrace{1}_{i^{\\text{th}}}, 0, \\ldots)$,\n$E_ne_i=e_i$ if $i\\leq2n$ and $E_ne_{2k-1}=E_ne_{2k}=\\frac\n{1}{2}(e_{2k-1}+e_{2k})$ if $n$ where $u_i$ denotes the $i$th row of U and $v_j$ denotes the $j$th column of V.\nThe solution of U,V can be found by solving the following equation \\eqref{lsm}:\n\\begin{equation}\n \\min_{U,V}=\\frac{1}{M}\\sum_{i=1}^{n}\\sum_{j=1}^{m}(r_{ij}-\\left)^2+\\lambda||U||^2_2+\\mu||V||^2_2\n \\label{lsm}\n\\end{equation}\n\nwhere $\\lambda$ and $\\mu$ are the regularization parameters. And we can solve the above problem by stochastic gradient descent with the following equations\\cite{koren2009matrix}:\n\\begin{equation}\n u_i^{new} = u_i^{old}-\\alpha\\Delta_{u_i}F(U^{old},V^{old})\n \\label{sgd1}\n\\end{equation}\n\\begin{equation}\n v_j^{new} = v_j^{old}-\\alpha\\Delta_{v_j}F(U^{old},V^{old})\n \\label{sgd2}\n\\end{equation}\nwhere\n\\begin{equation}\n \\Delta_{u_i}F(U,V) = -2\\sum_{j=1}^{m}v_j(r_{ij}-\\left)+2\\lambda u_i\n \\label{sgd3}\n\\end{equation}\n\\begin{equation}\n \\Delta_{v_j}F(U,V) = -2\\sum_{i=1}^{n}u_i(r_{ij}-\\left)+2\\lambda v_j\n \\label{sgd4}\n\\end{equation}\nThe parameters are updated iteratively until the model loss function is less than a fixed threshold or the gradient difference between the two iterations is small, then the model can be considered as convergent.\n\\subsection{Distributed recommendation system}\nWe suppose that in a distributed environment, items are shared but different users belong to different data sources. For example, for users who buy iPhones, some users may purchase through Taobao, while others will buy on the official website for quality reasons. \nDifferent users will rate the same item on different data sources. We assume that there are T data sources, the distributed matrix factorization recommendation system can be represented by Algorithm\\ref{alg:DMF}.\n\n\\begin{algorithm}\n \\caption{Distributed Matrix Factorization}\n \\label{alg:DMF}\n \\begin{algorithmic}\n \\REQUIRE $U_1,U_2,..,U_T,V,\\delta$\n \\STATE data sources init their user profile matrix $U_t$\n \\STATE server init item profile matrix $V,\\delta$\n \\REPEAT \n \\STATE \\textbf{data sources update:}\n \\FOR {$t=1;t<=T;t++$}\n \\STATE $\\Delta_{U_t}F(U,V)=-2(R_{U_t}-U_tV)V^T+2\\mu U_t$\n \\STATE $U_t^{new} = U_t^{old}-\\alpha\\Delta_{U_t}F(U,V)$\n \\STATE $Graident_t= -2U^T(R_{U_t}-U_tV)+2\\mu V$\n \\STATE send $Graident_t$ to server\n \\ENDFOR\n \\STATE \\textbf{server update:}\n \\STATE receive $Graidents$ from data sources\n \\STATE $G = \\sum_{t=1}^{T}Graident_t$\n \\STATE $V^{new} = V^{old} - G$\n \\UNTIL{$G<\\delta$}\n \\end{algorithmic}\n\\end{algorithm}\n\nUnder this framework, each data source holds its user profile matrix and keeps it secret to the outside. \nthe pubilc item profile matrix is stored in the central server. Each party uses the local rating matrix to update the user parameters, and only exposes the gradient of the item matrix to the server. \nThe server updates the item matrix after summarizing the gradient. This method only involves the transmission of gradients, therefore the security of local data is protected. \nHowever,transmit gradient can also exposes privacy. knowing the gradients of a data source uploaded in two continuous steps, it can infer the rating information by the equations \\eqref{leakage1}\\eqref{leakage2}. And see more detail in the paper\\cite{chai2019secure}. Therefore this article introduces secret sharing method in the transmission of graident, which make gradient transmission more efficient and safe.\n\n\\begin{equation}\n u_i^t=(r_{ij}-\\left< u_i^t,v_j^t\\right>)=G_j^t\n \\label{leakage1}\n\\end{equation}\n\n\\begin{equation}\n r_{ij}=\\frac{G_{jk}^t}{u_{ik}^t}+\\sum_{m=1}^{D}u_{im}^t v_{jm}^t\n \\label{leakage2}\n\\end{equation}\n\\subsection{Secret sharing}\nThe idea of secret sharing is to split the secret in an appropriate way, and each share after splitting is managed by different participants. A single participant cannot recover the secret information, and only several participants can cooperate to recover the secret message.\n\nThe figure\\ref{ss} gives a simple example of how to use sercet sharing. Two data sources own the number $X$ and $Y$ respectively, the server want to know the sum $X+Y$ but it will know nothing about X and Y. The process can be described as follows: \nfirstly, the original data is decomposed into two sub parts, and one sub part is exchanged between the two sides, and then the sum of the remaining sub parts with the part from other side is calculated. Finally, the solution of the original problem is obtained by summarizing the calculated sum. \nIn the process, the original data will not be exposed, so the sum operation can be completed under the premise of protecting data privacy. In addition, the multiplication can be realized by setting additional triples. In\\cite{zheng2020industrial}, the author uses secret sharing technology to implement multi-source federated neural network.\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/secret.png}}\n \\caption{An example of Secret sharing}\n \\label{ss}\n\\end{figure}\n\\subsection{Put all together}\nIn order to solve the privacy problem that may be caused by the exposure gradient, we propose a shared matrix factorization (SMF) method based on secret sharing. \nAs shown in the algorithm\\ref{alg:SMF}, the data source calculates the local user profile matrix parameters and the item matrix gradients are encrypted by secret sharing technology before transmitting to the server, and finally the encrypted gradients are summarized on the server to update the item profile matrix parameters.\n\\begin{algorithm}\n \\caption{Shared Matrix Factorization}\n \\label{alg:SMF}\n \\begin{algorithmic}\n \\REQUIRE $U_1,U_2,..,U_T,V,\\delta$\n \\STATE all parties initialize related parameters\n \\REPEAT \n \\STATE \\textbf{data sources update:}\n \\FOR {$t=1;t<=T;t++$}\n \\STATE update user profile matrix $U_t$\n \\STATE compute item matrix gradient $g_t^{plain}$\n \\STATE generate random number that meets $g_t^{plain}=g_{t}^{sub_1}+g_{t}^{sub_2}+..+g_{t}^{sub_T}$\n \\STATE keep $g_t^{sub_t}$ and send the rest to other data \n \\STATE receive $g^{sub_t}$ from others \n \\STATE compute hybrid gradient $g_t^{hybrid}=\\sum_{i=1}^T g_{i}^{sub_t}$\n \\STATE send hybrid gradient to server\n \\ENDFOR\n \\STATE \\textbf{server update:}\n \\STATE receive $g^{hybrid}$ from data sources\n \\STATE $G = \\sum_{t=1}^{T}g_t^{hybrid}$\n \\STATE $V^{new} = V^{old} - G$\n \\UNTIL{$G<\\delta$}\n \\end{algorithmic}\n\\end{algorithm}\n\n\\section{Evaluation}\n\\subsection{Dataset}\nTo make the recommendation algorithm be better applied to the actual scene, we choose the real world dataset Movielens, which has been applied in many recommendation systems, such as caser\\cite{tang2018personalized}, h4mf\\cite{wang2018modeling}. \nWe disorganize the rating matrix and randomly sampled the train\/test set according to the ratio of 7:3.\n\\subsection{Parameters}\nThrough training experience and super parameter adjustment, we choose a group of better parameter combinations, in which the profile matrix dimension $k = 100$,\nthe regularization parameters $reg_u=10^{-3}, reg_v=10^{-3}$, and the learning rate is $lr=10^{-2}$\n\\subsection{Environment}\nAll experiments are performed on a server with 2.5GHz 16-core CPU and 64GB RAM, where the operation system is Linux and the program language is Python. \nWe use multithreading to simulate multi-source data holder. And they communicate and exchange data through grpc. Each source will start a rpc server client to receive data from other clients \n\\subsection{Performance}\n\\paragraph{\\textbf{local and distributed comparison}}\nFirst, we tested the improvement that the distributed recommendation system can bring. We used the data provided by only local data, three data sources and five data sources. \nFor each additional data source, the number of rating users increased by 200, and the total number of movies remained at 500. The experimental results are shown in the figure\\ref{ld}. With the increase of data sources, the loss of the model decreases. \nThis is due to the increase in the number of users, the rating matrix is more perfect, which makes the item vector fitting better.\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/local_distribute.jpg}}\n \\caption{local and distributed recommender system comparison}\n \\label{ld}\n\\end{figure}\n\n\\paragraph{\\textbf{Horizontal comparison}}\nWe have tested the improvement brought by distributed recommendation. In federated learning, the main reason that affects the performance of distributed algorithms is the overhead of encryption methods. \nTherefore, we test the different performance between our algorithm and that without encryption. Since the main cost of secret sharing lies in the communication and exchange of sub secrets between nodes, we set different number of data sources for horizontal comparison. \nThe result is as shown in the figure\\ref{hc}. Compared with matrix factorization, the communication cost caused by secret sharing is less than the computation cost by matrix factorization. Therefore, the performance of shared MF is basically the same as that of common distributed recommendation system, which means our algorithm has strong practicability.\n\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=0.40\\textwidth]{mat\/raw_ss.jpg}}\n \\caption{time consumption with different data source numbers}\n \\label{hc}\n\\end{figure}\n\\paragraph{\\textbf{Vertical comparison}}\nIn the previous horizontal comparison, we studied the communication overhead caused by increasing data sources. In the process of secret sharing of each data source, the amount of data transmitted is determined by the size of the item profile matrix. \nTherefore, we select the appropriate number of data sources and set different number of items to test the algorithm performance. The experimental results are shown in the figure\\ref{vc}. There are three data sources on the left and five data sources on the right. \nIt is obvious that with the increase of the number of objects, the communication overhead does not increase significantly, which proves our algorithm is also very adaptable to large-scale items.\n\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=0.45\\textwidth]{mat\/items.jpg}}\n \\caption{time consumption with different item numbers}\n \\label{vc}\n\\end{figure}\n\\paragraph{\\textbf{why not homomorphic encryption}}\nFrom the perspective of cryptography, homomorphic encryption can guarantee zero leakage of data privacy. Therefore, the distributed recommendation system using this method has the best security in theory. \n\nHowever, the disadvantage of homomorphic encryption is very obvious. The computational cost of data encryption and decryption process is very high. We compared our algorithm with FedML which uses an addition Encryption Paillier and tested the time cost under the same condition. \n\nFrom the table\\ref{tab1}, we can see that homomorphic encryption scheme can work when the amount of data is small, but with the increase of data volume, the encryption time is obviously too high, which can not adapt to the actual large-scale recommendation scenarios.\n\n\\begin{table}[htbp]\n \\caption{SharedMF vs FedML}\n \\begin{center}\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n train time(sec)&items50&items200 &items500 \\\\\n \\hline\n FedML& 223.49 & 843.21 & 2064.62 \\\\\n \\hline\n SharedMF& 100.58 & 284.13 & 583.37 \\\\\n \\hline\n \\end{tabular}\n \\label{tab1}\n \\end{center}\n\\end{table}\n\n\n\\section{Concluson and futrue work}\nIn this paper, we propose a secure distributed matrix factorization recommendation system framework, called SharedMF. Specifically, we first construct a distributed recommendation scenario, and store user data and item information separately in the clients and a server. \nThe model is fitting by exchanging gradients between them, and the secret sharing technology is used to ensure the data privacy and security in the training process.\n\nIn the experimental stage, we first prove the usefulness of the distributed system to improve the accuracy of recommendation scenarios, and then compare the performance differences between our algorithm and the non-encrypted distributed recommendation to verify the practicability of the algorithm. \nMoreover, we test the existing solutions based on homomorphic encryption, which proves that our scheme is more robust to the increase in the number of users and items, and is more suitable for large-scale recommendation scenarios.\n\nWith the importance of privacy protection in recommendation system and machine learning increasing, federated learning technology based on cryptography is bound to be widely used. \nThe secret sharing technology used in this paper skilfully avoids the high computational complexity of traditional homomorphic encryption algorithm, and effectively improves the performance of privacy protection algorithm. However, it is worth mentioning that in this paper, secret sharing is only used to solve the privacy problem in the traditional algorithm matrix factorization. \nHow to apply it in the current popular deep neural network will be our further research topic.\n\n\n\\bibliographystyle{mat\/IEEEtran.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nThe main notions in nonrelativistic quantum mechanics are the \\Sh\\\nand wave function $|\\psi\\rangle$. The density matrix is an\nartificial construction, which, as will be shown bellow, can be\ncontradictory. We will consider the simplest case of the density\nmatrix, describing a monochromatic nonpolarized neutron beam.\n\nA monochromatic non polarized neutron beam is characterized by the\ndensity matrix\n\\begin{equation}\\label{dm}\n\\rho=\\fr1{2}\\Big(|u\\rangle\\langle u|+|d\\rangle\\langle d|\\Big),\n\\end{equation}\nwhich is one half of the unit matrix. The states $|u,d\\rangle$\ncorrespond to wave functions for neutrons polarized along and\nopposite some direction, which is known as quantization axis. The\nchoice of the quantization axis, however, is not important,\nbecause the density matrix \\eref{dm} is invariant with respect to\nsuch a choice. Indeed, if one chooses the quantization axis along\nsome unit vector $\\av$, then the matrix \\eref{dm} becomes\n\\begin{equation}\\label{2}\n\\rho=\\fr1{2}\\Big(|\\av\\rangle\\langle \\av|+|-\\av\\rangle\\langle\n-\\av|\\Big).\n\\end{equation}\nIf one chooses another axis $\\bb$, then, since\n\\begin{equation}\\label{3}\n|\\av\\rangle=\\alpha|\\bb\\rangle+\\beta|-\\bb\\rangle,\\qquad\n|-\\av\\rangle=\\alpha^*|-\\bb\\rangle-\\beta^*|\\bb\\rangle,\n\\end{equation}\nwhere $|\\alpha|^2+|\\beta|^2=1$, one obtains\n\\begin{equation}\\label{4}\n\\rho=\\fr1{2}\\Big(\\lt[\\alpha|\\bb\\rangle+\\beta|-\\bb\\rangle\\rt]\\lt[\\alpha^*\\langle\\bb|+\\beta^*\\langle-\\bb|\\rt]\n+$$\n$$+\\lt[\\beta^*|\\bb\\rangle-\\alpha^*|-\\bb\\rangle\\rt]\\lt[\\beta\\langle\\bb|-\\alpha\\langle-\\bb|\\rt]\\Big)=$$\n$$=\n\\fr1{2}\\Big(|\\bb\\rangle\\langle \\bb|+|-\\bb\\rangle\\langle\n-\\bb|\\Big).\n\\end{equation}\nFor instance, if $\\av$ is along $y$ axis, and $\\bb$ is along\n$z$-axis, one has\n\\begin{equation}\\label{5}\n|y\\rangle=\\fr1{\\sqrt2}{1\\choose\ni}=\\fr1{\\sqrt2}\\lt(|z\\rangle+i|-z\\rangle\\rt),\\qquad\n|-y\\rangle=\\fr1{\\sqrt2}{i\\choose1}=\\fr1{\\sqrt2}\\lt[|-z\\rangle+i|z\\rangle\\rt],\n\\end{equation}\nand\n\\begin{equation}\\label{6}\n\\rho=\\fr1{2}\\Big(|+z\\rangle\\langle+z|+|-z\\rangle\\langle-z|\\Big)=\\fr1{2}\\Big(|+y\\rangle\\langle+y|+|-y\\rangle\\langle-y|\\Big).\n\\end{equation}\nSo two axes are equivalent for the density matrix. However these\naxes can be discriminated by an experimental equipment, and our\ngoal is to show how it is possible. To achieve it let's first show\nhow one can find polarization direction of a polarized beam.\n\n\\section{A method for polarization direction measurement}\n\nThe principle is based on an effect known in neutron\noptics~~\\cite{vf,ga,uig}, and is related to spin flip with the\nhelp of a resonant radio frequency (rf) spin-flipper. Such a\nspin-flipper is a coil with a permanent magnetic field $\\B_0$ and\nperpendicular to it rotating counterclockwise rf-field\n\\begin{equation}\\label{rf}\n\\B_{rf}=b\\Big(\\cos(\\omega t),\\sin(\\omega t).0\\Big),\n\\end{equation}\nwhere $\\omega=2\\mu B_0\/\\hbar$, and $\\mu$ is magnetic moment of the\nneutron, which is aligned oppositely to the neutron spin $\\sbb$.\nDirection of $\\B_0$ can be accepted as the quantization z-axis.\nInteraction of neutrons with such a flipper can be solved exactly\nand analytically, and the solution can be explained as\nfollows~\\cite{uig}.\n\nThe neutron interaction with magnetic field is described by the\npotential $-\\mb\\cdot\\B_0$. Therefore neutrons in the state\n$|z\\rangle$ entering the field $\\B_0$ are decelerated because the\nfield in this case creates a potential barrier of height $\\mu\nB_0$.\n\nInside the flipper the rf-field turns the spin down, i.e.\ntransforms the state $|z\\rangle$ into $|-z\\rangle$. In this state\nthe interaction $-\\mb\\cdot\\B_0$ becomes negative, so the potential\nbarrier transforms into potential well of depth $\\mu B_0$.\nTherefore after exit from the flipper and its magnetic field\n$\\B_0$ the neutron decelerates once again. In total the neutron\nenergy after transmission through the spin flipper decreases by\namount $2\\mu B_0$, which means emission of an rf quantum:\n$\\hbar\\omega=2\\mu B_0$. The wave functions before and after spin\nflipper are\n\\begin{equation}\\label{rf2}\n|\\psi_{in}(x,t)\\rangle=\\exp(ikx-i\\Omega t)|z\\rangle,\n\\end{equation}\n\\begin{equation}\\label{rf3}\n|\\psi_{out}(x,t)\\rangle=\\exp(ik_-(x-D)-i(\\Omega-\\omega)\nt)|-z\\rangle,\n\\end{equation}\nrespectively. Here $x$ is the axis of propagation, $D$ is\nthickness of the spin-flipper, $k$ is initial wave number,\n$\\Omega=\\hbar k^2\/2m$, $m$ is the neutron mass, and\n$k_-=\\sqrt{k^2-2m\\omega\/\\hbar}$. If the incident neutron has the\nstate $|-z\\rangle$ it accelerates, and after spin-flipper has\nenergy larger than original one by the amount $2\\mu B_0$, which\nmeans absorbtion of an rf quantum: $\\hbar\\omega=2\\mu B_0$. The\nwave functions before and after spin flipper in this case are\nrespectively\n\\begin{equation}\\label{rf4}\n|\\psi_{in}(x,t)\\rangle=\\exp(ikx-i\\Omega t)|-z\\rangle,\n\\end{equation}\n\\begin{equation}\\label{rf5}\n|\\psi_{out}(x,t)\\rangle=\\exp(ik_+(x-D)-i(\\Omega+\\omega)\nt)|z\\rangle,\n\\end{equation}\nwhere $k_+=\\sqrt{k^2+2m\\omega\/\\hbar}$.\n\nIf the incident neutron has a polarization\n$|\\xi\\rangle=\\alpha|z\\rangle+\\beta|-z\\rangle$, its wave function\nbefore and after spin flipper are respectively\n\\begin{equation}\\label{rf6}\n|\\psi_{in}(x,t)\\rangle=\\exp(ikx-i\\Omega t)(\\alpha|z\\rangle+\\beta|-z\\rangle),\n\\end{equation}\n\\begin{equation}\\label{rf7}\n|\\psi_{out}(x,t)\\rangle=\\alpha\\exp(ik_-(x-D)-i(\\Omega-\\omega)\nt)|-z\\rangle+$$\n$$+\\beta\n\\exp(ik_+(x-D)-i(\\Omega+\\omega) t)|z\\rangle.\n\\end{equation}\nThe spin arrow of this\nstate represents a rotating spin wave propagating along $x$-axis.\n\nLet's put at some position $x=x_0$ an analyzer, which transmits\nonly neutrons polarized along $y$-axis. Since\n\\begin{equation}\\label{5}\n|+z\\rangle=\\fr1{\\sqrt2}(|+y\\rangle-i|-y\\rangle),\\qquad\n|-z\\rangle=\\fr 1{i\\sqrt2}(|+y\\rangle+i|-y\\rangle),\n\\end{equation}\nwhere $|\\pm y\\rangle$ denote states with polarization along and\nopposite $y$ axis, the neutron state \\eref{rf7} after the analyzer\nis\n\\begin{equation}\\label{6}\n|\\psi_{+y}(x_0,t)\\rangle=$$ $$\\fr {|+y\\rangle}{i\\sqrt2}\\lt(\\alpha\ne^{ ik_-(x_0-D)-i(\\Omega-\\omega)t}+i\\beta e^{\nik_+(x_0-D)-i(\\Omega+\\omega)t}\\rt),\n\\end{equation}\nand intensity of the neutron beam after the analyzer at some\nposition $x_0$ is\n\\begin{equation}\\label{7}\nI_{+y}(x_0,t)=\\fr12\\lt[|\\alpha|^2+|\\beta|^2+2|\\alpha\\beta|\\cos(\\varphi+2\\omega\nt)\\rt],\\end{equation}\n where $\\varphi$ is some phase. We see that\n the beam has density modulation with time, and visibility of the modulation\n \\begin{equation}\\label{8}\n V=\\fr{2|\\alpha\\beta|}{|\\alpha|^2+|\\beta|^2}=\\fr{2|\\alpha\/\\beta|}{1+|\\alpha|^2\/|\\beta|^2}\n\\end{equation}\ndetermines ratio $|\\alpha\/\\beta|$ and, therefore, the polar angle\nof the incident neutron spin arrow with respect to $z$-axis. If\n$\\alpha$ or $\\beta$ are zero, i.e. incident neutron is polarized\nalong or opposite spin-flipper axis, oscillations are absent.\n\n\\section{An experimental possibility for discrimination between $z$ and $y$ quantization axes}\n\nNow let's suppose that quantization axis is directed along\n$y$-axis. It means that the number $N_+$ of particles in the state\n$|+y\\rangle$ is the same as the number $N_-$ in the state\n$|-y\\rangle$. Since $|\\pm y\\rangle=(|\\pm z\\rangle+ i|\\mp\nz\\rangle)\/\\sqrt2$, we have according to \\eref{6} the intensities\nafter $y$-analyzer for two incident components $|\\pm y\\rangle$\nmeasured by a detector at some position $x_0$ to be\n\\begin{equation}\\label{7a}\nI_{+y}^\\pm(x_0,t)=\\fr{N_\\pm}2\\lt[1\\pm\\cos(2\\omega\nt)\\rt],\\end{equation} where upper index points out what was the\nincident component, and for simplicity we put the phase $\\varphi$\nin \\eref{7a} to zero, because it is the same for all the\nparticles.\n\nThe sum of averaged over time two intensities is a constant\n\\begin{equation}\\label{7a1}\n\\langle I_{+y}(t)\\rangle=\\langle I^+_{+y}(t)\\rangle+\\langle\nI^-_{+y}(t)\\rangle=$$ $$=\\fr{\\langle\nN_+\\rangle}2\\lt[1+\\cos(2\\omega t)\\rt]+\\fr{\\langle\nN_-\\rangle}2\\lt[1-\\cos(2\\omega t)\\rt]=N_0,\\end{equation} where\n$N_0=\\langle N_+\\rangle=\\langle N_-\\rangle$.\n\nHowever besides the average value there are also fluctuations of\nneutron count rate. We can naturally suppose that the fluctuations\nof two incident spin components are independent, and obey the\nPoisson statistics. Then fluctuations of neutron flux density\nafter $y$-analyzer will be\n\n\\begin{equation}\\label{7a2}\n\\langle|\\delta I_{+y}(t)|^2\\rangle=\\langle|\\delta\nI^+_{+y}(t)|^2\\rangle+\\langle|\\delta I^-_{+y}(t)|^2\\rangle=$$\n$$=\\Big\\langle\\fr{\\delta N_+}2\\lt[1+\\cos(2\\omega\nt)\\rt]\\Big\\rangle^2+\\Big\\langle\\fr{\\delta N_-}2\\lt[1-\\cos(2\\omega\nt)\\rt]\\Big\\rangle^2=\\fr{N_0}{2}(1+\\cos^2(2\\omega\nt)).\\end{equation}\n\nTo see these oscillations one should divide the period\n$T=\\pi\/2\\omega$ over $N$ small intervals $\\Delta T=T\/N$ and sum\nthe value\n\\begin{equation}\\label{7aa2}\n\\fr{\\langle|\\delta\nI_{+y}(t_n)|^2\\rangle}{N_0}=\\fr{1}2\\lt[1+\\cos^2(t_n\/\nT)\\rt],\\end{equation} at $t_n=n\\Delta T$ over many periods $T$.\n\nThis way one can discriminate between two quantization axes $z$,\nand $y$. Therefore these quantization axes are not equivalent,\nwhereas according to density matrix expression they are absolutely\nequivalent. This is the contradiction we wanted to point to.\n\n\\section{Conclusion}\n\nThe main element of \\qm\\ is a wave function, and corresponding to\nit a pure state. If one has an ensemble of particles with\ndifferent pure states, and the distribution of different states is\ncharacterized by probabilities, one must calculate a process with\npure states and then average over probabilities. This is the way\nneutron scattering cross sections are calculated. First they are\ncalculated for a pure state of an incident plain wave, and then\nthe obtained cross section is averaged over probability\ndistribution of the incident plain waves. Of course the density\nmatrix also can be useful, but because of discovered\ncontradiction, one must be very careful with it.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\n\nState-based models of concurrent systems are standardly considered\nunder a wide range of system equivalences, typically located between\ntwo extremes respectively representing \\emph{linear time} and\n\\emph{branching time} views of system evolution. Over labelled\ntransition systems, one specifically has the well-known \\emph{linear\n time -- branching time spectrum} of system equivalences between\ntrace equivalence and\nbisimilarity~\\cite{vanglabbeek2001linear}. Similarly, e.g.\\\nprobabilistic automata have been equipped with various semantics\nincluding strong bisimilarity~\\cite{LarsenSkou91}, probabilistic\n(convex) bisimilarity~\\cite{SegalaLynch94}, and distribution\nbisimilarity (e.g.~\\cite{DengEA08,DoyenEA08}). New equivalences keep\nappearing in the literature, e.g.~for non-deterministic probabilistic\nsystems~\\cite{BonchiEA19,vanHeerdtEA18}.\n\nThis motivates the search for unifying principles that allow for a\ngeneric treatment of process equivalences of varying degrees of\ngranularity and for systems of different branching types\n(non-deterministic, probabilistic etc.). As regards the variation of\nthe branching type, universal coalgebra~\\cite{Rutten00} has emerged as\na widely-used uniform framework for state-based systems covering a\nbroad range of branching types including besides non-deterministic and\nprobabilistic, or more generally weighted, branching also, e.g.,\nalternating, neighbourhood-based, or game-based systems. It is based\non modelling the system type as an endofunctor on some base category,\noften the category of sets.\n\nUnified treatments of system equivalences, on the other hand, are so\nfar less well-established, and their applicability is often more\nrestricted. Existing approaches include coalgebraic trace semantics in\nKleisli~\\cite{HasuoEA07} and Eilenberg-Moore\ncategories~\\cite{KissigKurz10,JacobsEA12,sbbr13,bms13,BonchiEA19,vanHeerdtEA18},\nrespectively. Both semantics are based on decomposing the coalgebraic\ntype functor into a monad, the \\emph{branching type}, and a functor,\nthe \\emph{transition type} (in different orders), and require suitable\ndistributive laws between these parts; correspondingly, they grow\nnaturally out of the functor but on the other hand apply only to\nfunctors that admit the respective form of decomposition. In the\npresent work, we build on a more general approach introduced by\nPattinson and two of us, based on mapping the coalgebraic type functor\ninto a \\emph{graded monad}~\\cite{MiliusEA15}. Graded monads correspond\nto algebraic theories where operations come with an explicit notion of\n\\emph{depth}, and allow for a stepwise evaluation of process\nsemantics. Maybe most notably, graded monads systematically support a\nreasonable notion of \\emph{graded logic} where modalities are\ninterpreted as \\emph{graded algebras} for the given graded monad. This\napproach applies to all cases covered in the mentioned previous\nframeworks, and additional cases that do not fit any of the earlier\nsetups. We emphasize that graded monads are geared towards\n\\emph{inductively} defined equivalences such as finite trace semantics\nand finite-depth bisimilarity; we leave a similarly general treatment\nof infinite-depth equivalences such as infinite trace equivalence and\nunbounded-depth bisimilarity to future work. To avoid excessive\nverbosity, we restrict to models with finite branching\nthroughout. Under finite branching, finite-depth equivalences\ntypically coincide with their infinite-depth counterparts, e.g.\\\nstates of finitely branching labelled transition systems are bisimilar\niff they are finite-depth bisimilar, and infinite-trace equivalent iff\nthey are finite-trace equivalent.\n\nOur goal in the present work is to illustrate the level of generality\nachievable by means of graded monads in the dimension of system\nequivalences. We thus pick a fixed coalgebraic type, that of labelled\ntransition systems, and elaborate how a number of equivalences from\nthe linear time -- branching time spectrum are cast as graded\nmonads. In the process, we demonstrate how to extract logical\ncharacterizations of the respective equivalences from most of the\ngiven graded monads. For the time being, none of the logics we find\nare sensationally new, and in fact van Glabbeek already provides\nlogical characterizations in his exposition of the linear time --\nbranching time spectrum~\\cite{vanglabbeek2001linear}; an overview of\ncharacteristic logics for non-deterministic and probabilistic\nequivalences is given by Bernardo and\nBotta~\\cite{bernardo-botta:characterising-logics}. The emphasis in the\nexamples is mainly on showing how the respective logics are developed\nuniformly from general principles.\n\nUsing these examples as a backdrop, we develop the theory of graded\nmonads and graded logics further. In particular,\n\\begin{itemize}\n\\item we give a more economical characterization of depth-$1$ graded\n monads involving only two functors (rather than an infinite sequence\n of functors);\n\\item we extend the logical framework by a treatment of propositional\n operators -- previously regarded as integrated into the modalities\n -- as first class citizens; \n\\item we prove, as our main technical result, a generic expressiveness\n criterion for graded logics guaranteeing that inequivalent states are\n separated by a trace formula. \n\\end{itemize}\nOur expressiveness criterion subsumes, at the branching-time end of\nthe spectrum, the classical Hennessy-Milner\ntheorem~\\cite{HennessyMilner85} and its coalgebraic\ngeneralization~\\cite{Pattinson04,Schroder08} as well as expressiveness\nof probabilistic modal logic with only\nconjunction~\\cite{DesharnaisEA98}; we show that it also covers\nexpressiveness of the respective graded logics for more coarse-grained\nequivalences along the linear time -- branching time spectrum. To\nillustrate generality also in the branching type, we moreover provide\nan example in a probabilistic setting, specifically we apply our\nexpressiveness criterion to show expressiveness of a quantitative\nmodal logic for probabilistic trace equivalence.\n\n\\myparagraph{Related Work} Fahrenberg and\nLegay~\\cite{FahrenbergLegay17} characterize equivalences on the linear\ntime -- branching time spectrum by suitable classes of modal\ntransition systems. We have already mentioned previous work on\ncoalgebraic trace semantics in Kleisli and Eilenberg-Moore\ncategories~\\cite{HasuoEA07,KissigKurz10,JacobsEA12,sbbr13,bms13,BonchiEA19,vanHeerdtEA18}. A\ncommon feature of these approaches is that, more precisely speaking,\nthey model \\emph{language} semantics rather than trace semantics --\ni.e.\\ they work in settings with explicit successful termination, and\nconsider only successfully terminating traces. When we say that graded\nmonads apply to all scenarios covered by these approaches, we mean\nmore specifically that the respective language semantics are obtained\nby a further canonical quotienting of our trace\nsemantics~\\cite{MiliusEA15}. Having said that graded monads are\nstrictly more general than Kleisli and Eilenberg-Moore style trace\nsemantics, we hasten to add that the more specific setups have their\nown specific benefits including final coalgebra characterizations and,\nin the Eilenberg-Moore setting, generic determinization procedures. A\nfurther important piece of related work is Klin and Rot's method of\ndefining trace semantics via the choice of a particular flavour of\ntrace logic~\\cite{KlinRot15}. In a sense, this approach is opposite to\nours: A trace logic is posited, and then two states are declared\nequivalent if they satisfy the same trace formulae. In our approach\nvia graded monads, we instead pursue the ambition of first fixing a\nsemantic notion of equivalence, and then designing a logic that\ncharacterizes this equivalence. Like Klin and Rot, we view trace\nequivalence as an inductive notion, and in particular limit attention\nto finite traces; coalgebraic approaches to infinite traces exist, and\nmostly work within the Kleisli-style\nsetup~\\cite{Jacobs04,Cirstea11,KerstanKonig13,Cirstea14,Cirstea15,UrabeHasuo15,Cirstea17}. Jacobs,\nLevy and Rot~\\cite{JacobsEA18} use corecursive algebras to provide a\nunifying categorical view on the above-mentioned approaches to traces\nvia Kleisli- and Eilenberg-Moore categories and trace logics,\nrespectively. This framework does not appear to subsume the approach\nvia graded monads, and like for the previous approaches we are not\naware that it covers semantics from the linear time -- branching time\nspectrum other than the end points (bisimilarity and trace\nequivalence).\n\n\\section{Preliminaries: Coalgebra}\\label{sec:prelim}\n\nWe recall basic definitions and results in \\emph{(universal)\n coalgebra}~\\cite{Rutten00}, a framework for the unified treatment of\na wide range of reactive systems. We write~$1=\\{\\star\\}$ for a fixed\none-element set, and $!\\colon X\\to 1$ for the unique map from a set~$X$\ninto~$1$. We write $f\\cdot g$ for the composite of maps $g\\colon X\\to Y$,\n$f\\colon Y\\to Z$, and $\\langle f,g\\rangle\\colon X\\to Y\\times Z$ for the pair map\n$x\\mapsto(f(x),g(x))$ formed from maps $f\\colon X\\to Y$, $g\\colon X\\to Z$.\n\nCoalgebra encapsulates the branching type of a given species of\nsystems as a \\emph{functor}, for purposes of the present paper on the\ncategory of sets. Such a functor $G\\colon \\Set\\to\\Set$ assigns to each\nset~$X$ a set~$GX$, whose elements we think of as structured\ncollections over~$X$, and to each map $f\\colon X\\to Y$ a map $Gf\\colon\nGX\\to GY$,\npreserving identities and composition. E.g.\\ the \\emph{(covariant)\n powerset functor}~$\\mathcal{P}$ assigns to each set~$X$ the powerset\n$\\mathcal{P} X$ of~$X$, and to each map $f\\colon X\\to Y$ the map\n$\\mathcal{P} f\\colon \\mathcal{P} X\\to\\mathcal{P} Y$ that takes direct images. (We mostly omit\nthe description of the action of functors on maps in the sequel.)\nSystems with branching type described by~$G$ are then abstracted as\n\\emph{$G$-coalgebras}, i.e.\\ pairs $(X,\\gamma)$ consisting of a\nset~$X$ of \\emph{states} and a map $\\gamma\\colon X\\to GX$, the\n\\emph{transition map}, which assigns to each state $x\\in X$ a\nstructured collection $\\gamma(x)$ of successors. For instance, a\n$\\mathcal{P}$-coalgebra assigns to each state a set of successors, and thus\nis the same as a transition system.\n\\begin{example}\\label{expl:coalg}\n \\begin{longitemslist\n \\item Fix a set~$\\mathcal{A}$ of \\emph{actions}. The functor\n $\\mathcal{A}\\times(-)$ assigns to each set $X$ the set $\\mathcal{A}\\times X$;\n composing this functor with the powerset functor, we obtain the\n functor $G=\\mathcal{P}(\\mathcal{A}\\times(-))$ whose coalgebras are precisely\n labelled transition systems (LTS): A $G$-coalgebra assigns to each\n state~$x$ a set of pairs $(\\sigma,y)$, indicating that~$y$ is a\n successor of~$x$ under the action~$\\sigma$.\n \\item The \\emph{(finite) distribution functor}~$\\mathcal{D}$ maps a\n set~$X$ to the set of finitely supported discrete probability\n distributions on~$X$. These can be represented as probability mass\n functions $\\mu\\colon X\\to[0,1]$, with $\\sum_{x\\in X}\\mu(x)=1$ and with\n the \\emph{support} $\\{x\\in X\\mid \\mu(x)>0\\}$ being\n finite. Coalgebras for~$\\mathcal{D}$ are precisely Markov\n chains. Composing with $\\mathcal{A}\\times(-)$ as above, we obtain the\n functor $\\mathcal{D}(\\mathcal{A}\\times(-))$, whose coalgebras are\n \\emph{generative probabilistic transition systems}, i.e.~assign\n to each state a distribution over pairs consisting of an action\n and a successor state. \n \\end{longitemslist}\n\\end{example}\nAs indicated in the introduction, we restrict attention to\n\\emph{finitary} functors~$G$, in which every element $t\\in GX$ is\nrepresented using only finitely many elements of~$X$; formally, each\nset~$GX$ is the union of all sets $Gi_Y[GY]$ where $Y$ ranges over\nfinite subsets of~$X$ and $i_Y$ denotes the injection $i_Y\\colon Y\\hookrightarrow X$.\nConcretely, this means that we restrict the set~$\\mathcal{A}$ of actions to\nbe finite, and work with the \\emph{finite powerset functor}~$\\Pow_\\omega$\n(which maps a set~$X$ to the set of its finite subsets) in lieu\nof~$\\mathcal{P}$. ($\\mathcal{D}$ as defined above is already finitary.)\n\nCoalgebra comes with a natural notion of \\emph{behavioural\n equivalence} of states. A \\emph{morphism}\n$f\\colon (X,\\gamma)\\to(Y,\\delta)$ of $G$-coalgebras is a map $f\\colon X\\to Y$ that\ncommutes with the transition maps, i.e.\\\n$\\delta\\cdot f=Gf\\cdot\\gamma$. Such a morphism is seen as preserving\nthe behaviour of states (that is, behaviour is defined as being\nwhatever is preserved under morphisms), and consequently states\n$x\\in X$, $z\\in Z$ in coalgebras $(X,\\gamma)$, $(Z,\\zeta)$ are\n\\emph{behaviourally equivalent} if there exist coalgebra morphisms\n$f\\colon (X,\\gamma)\\to(Y,\\delta)$, $g\\colon (Z,\\zeta)\\to(Y,\\delta)$ such that\n$f(x)=g(z)$. For instance, states in LTSs are\nbehaviourally equivalent iff they are bisimilar in the standard sense,\nand similarly, behavioural equivalence on generative probabilistic\ntransition systems coincides with the standard notion of probabilistic\nbisimilarity~\\cite{Klin09}. We have an alternative notion of\nfinite-depth behavioural equivalence: Given a $G$-coalgebra\n$(X,\\gamma)$, we define a series of maps $\\gamma_n\\colon X\\to G^n1$\ninductively by taking $\\gamma_0$ to be the unique map $X\\to 1$, and\n$\\gamma_{n+1} = G\\gamma_n \\cdot\\gamma$. (These are the first $\\omega$\nsteps of the \\emph{canonical cone} from~$X$ into the \\emph{final\n sequence} of~$G$~\\cite{AdamekKoubek77}.) Then states $x,y$ in\ncoalgebras $(X,\\gamma)$, $(Z,\\zeta)$ are \\emph{finite-depth\n behaviourally equivalent} if $\\gamma_n(x)=\\zeta_n(y)$ for all $n$;\nin the case where~$G$ is finitary, finite-depth behavioural equivalence\ncoincides with behavioural equivalence~\\cite{worrell}.\n\n\n\\section{Graded Monads and Graded Theories}\n\n\\noindent We proceed to recall background on system semantics via\ngraded monads introduced in our previous work~\\cite{MiliusEA15}. We\nformulate some of our results over general base categories~$\\mathbf{C}$,\nusing basic notions from category theory~\\cite{MacLane98,Pierce91};\nfor the understanding of the examples, it will suffice to think of\n$\\mathbf{C}=\\Set$. Graded monads were originally introduced by\nSmirnov~\\cite{smirnov08} (with grades in a commutative monoid, which\nwe instantiate to the natural numbers):\n\\begin{defn}[Graded Monads]\n A \\emph{graded monad}~$M$ on a category $\\cat C$ consists of a\n family of functors $(M_n\\colon \\cat C \\to \\cat C)_{n<\\omega}$, a natural\n transformation $\\ensuremath{\\eta}\\xspace\\colon \\Id \\to M_0$ (the \\emph{unit}) and a family\n of natural transformations \n \\iffull\n \\[\n \\ensuremath{\\mu}\\xspace^{nk}\\colon M_n M_k \\to M_{n+k}\\quad (n,k<\\omega)\n \\]\n \\else\\\/$\\ensuremath{\\mu}\\xspace^{nk}\\colon M_n M_k \\to M_{n+k}$ for $n,k<\\omega$, \\fi\n (the \\emph{multiplication}), satisfying the \\emph{unit laws},\n $\\ensuremath{\\mu}\\xspace^{0n}\\cdot\\ensuremath{\\eta}\\xspace M_n = \\id_{M_n} = \\ensuremath{\\mu}\\xspace^{n0}\\cdot M_n\\ensuremath{\\eta}\\xspace$,\n for all $n<\\omega$, and the \\emph{associative law}\n \\iffull\n \\[\n \\begin{tikzcd}\n M_nM_kM_m\\ar{d}{\\ensuremath{\\mu}\\xspace^{nk}M_m}\\ar{r}{M_n\\ensuremath{\\mu}\\xspace^{km}} & M_nM_{k+m}\\ar{d}{\\ensuremath{\\mu}\\xspace^{n,k+m}} \\\\\n M_{n+k}M_m\\ar{r}{\\ensuremath{\\mu}\\xspace^{n+k,m}} & M_{n+k+m}\n \\end{tikzcd}\n \\qquad\\text{for all $k,n,m<\\omega$.}\n \\]\n \\else\\\/$\\ensuremath{\\mu}\\xspace^{n,k+m} \\cdot M_n \\ensuremath{\\mu}\\xspace^{km} = \\ensuremath{\\mu}\\xspace^{n+k,m} \\cdot\n \\ensuremath{\\mu}\\xspace^{nk}M_m$ for all $k,n,m<\\omega$.\\fi\n\\end{defn}\nNote that it follows that $(M_0, \\eta, \\mu^{00})$ is a (plain)\nmonad. For $\\cat C = \\Set$, the standard equivalent presentation of\nmonads as algebraic theories carries over to graded monads. Whereas\nfor a monad $T$, the set $TX$ consists of terms over $X$ modulo\nequations of the corresponding algebraic theory, for a graded monad\n$(M_n)_{n<\\omega}$, $M_nX$ consists of terms of uniform depth $n$\nmodulo equations:g\n\n\\begin{defn}[Graded Theories~\\cite{MiliusEA15}]\n A \\emph{graded theory} $(\\Sigma,E,d)$ consists of an algebraic\n theory, i.e.\\ a (possibly class-sized and infinitary) algebraic\n signature $\\Sigma$ and a class $E$ of equations, and an assignment\n $d$ of a \\emph{depth} $d(f)<\\omega$ to every operation symbol\n $f\\in\\Sigma$. This induces a notion of a term \\emph{having uniform\n depth $n$}: all variables have uniform depth $0$, and\n $f(t_1,\\dots,t_n)$ with $d(f)=k$ has uniform depth $n+k$ if all\n $t_i$ have uniform depth $n$. (In particular, a constant $c$ has\n uniform depth $n$ for all $n\\ge d(c)$). We require that all\n equations $t=s$ in $E$ have uniform depth, i.e.\\ that both $t$ and\n $s$ have uniform depth~$n$ for some~$n$. Moreover, we require that\n for every set $X$ and every $k<\\omega$, the class of terms of\n uniform depth $k$ over variables from $X$ modulo provable equality\n is small (i.e.\\ in bijection with a set).\n\\end{defn}\n\\noindent Graded theories and graded monads on $\\Set$ are essentially\nequivalent concepts~\\cite{smirnov08,MiliusEA15}. In particular, a\ngraded theory $(\\Sigma,E,d)$ induces a graded monad~$M$ by taking\n$M_nX$ to be the set of $\\Sigma$-terms over $X$ of uniform depth~$n$,\nmodulo equality derivable under $E$.\n\n\\begin{example}\\label{E:graded-monad}\n We recall some examples of graded monads and theories~\\cite{MiliusEA15}.\n \\begin{longitemslist\n \\item\\label{E:graded-monad:bisim} For every endofunctor $F$ on\n $\\cat C$, the $n$-fold composition $M_n = F^n$ yields a graded\n monad with unit $\\eta = \\id_{\\Id}$ and $\\mu^{nk} = \\id_{F^{n+k}}$.\n\n \\item\\label{E:graded-monad:kleisli} As indicated in the\n introduction, distributive laws yield graded monads: Suppose that\n we are given a monad $(T,\\ensuremath{\\eta}\\xspace,\\ensuremath{\\mu}\\xspace)$, an endofunctor $F$ on\n $\\cat C$ and a distributive law of $F$ over $T$ (a so-called\n \\emph{Kleisli law}), i.e.\\ a natural transformation\n $\\lambda\\colon FT \\to TF$ such that $\\lambda \\cdot F\\eta = \\eta F$\n and $\\lambda \\cdot F\\mu = \\mu F \\cdot T\\lambda \\cdot \\lambda T$.\n Define natural transformations $\\lambda^n\\colon F^nT \\to TF^n$\n inductively by $\\lambda^0 = \\id_T$ and\n $\\lambda^{n+1} = \\lambda^{n}F \\cdot F^{n}\\lambda$. Then we obtain\n a graded monad with $M_n = TF^n$, unit $\\eta$, and multiplication\n $\\mu^{nk} = \\mu F^{n+1} \\cdot T\\lambda^n F^k$. The situation is\n similar for distributive laws of~$T$ over~$F$ (so-called\n \\emph{Eilenberg-Moore laws}).\n \\item\\label{E:graded-monad:tr} As a special case of\n \\ref{E:graded-monad:kleisli}., for every monad $(T, \\eta, \\mu)$ on\n $\\Set$ and every set $\\mathcal{A}$, we obtain a graded monad with\n $M_nX = T(\\mathcal{A}^n \\times X)$. Of particular interest to us will be\n the case where $T = \\Pow_\\omega$, which is generated by the algebraic\n theory of join semilattices (with bottom). The arising graded monad\n $M_n=\\Pow_\\omega(\\mathcal{A}^n\\times X)$, which is\n associated with trace equivalence, is generated by the graded\n theory consisting, at depth~$0$, of the operations and equations\n of join semilattices, and additionally a unary operation of\n depth~$1$ for each $\\sigma \\in \\mathcal{A}$, subject to (depth-$1$)\n equations expressing that these unary operations distribute over\n joins.\n \\end{longitemslist}\n\\end{example}\n\n\\myparagraph{Depth-1 Graded Monads and Theories}\nwhere operations and equations have depth at most~$1$ are a particularly convenient case for\npurposes of building algebras of graded monads; in the following, we elaborate on this\ncondition.\n\\begin{defn}[Depth-1 Graded\n Theory~\\cite{MiliusEA15}]\\label{D:d1}\n A graded theory is called \\emph{depth-$1$} if all its operations\n and equations have depth at most~$1$. A graded monad on $\\Set$ is\n \\emph{depth-1} if it can be generated by a depth-1 graded theory.\n\\end{defn}\n\\begin{proposition}[Depth-1 Graded Monads~\\cite{MiliusEA15}]\\label{P:d1}\n A graded monad $((M_n),\\eta,(\\mu^{nk}))$ on $\\Set$ is depth-$1$\n iff the diagram below is objectwise a coequalizer diagram in\n $\\Set^{M_0}$ for all $n<\\omega$:\n \\begin{equation}\\label{eq:mu1n}\n \\xymatrix@1{M_1M_0M_n \\ar@<3pt>[rr]^{M_1\\mu^{0n}} \n \\ar@<-3pt>[rr]_{\\mu^{10}M_n} && M_1 M_n \\ar[r]^{\\mu^{1n}} & M_{1+n}}.\n \\end{equation}\n\\end{proposition}\n\n\\begin{example}\\label{E:d1}\n All graded monads in \\autoref{E:graded-monad} are depth $1$:\n for~\\ref{E:graded-monad:bisim}., this is easy to see,\n for~\\ref{E:graded-monad:tr}., it follows from the presentation as a\n graded theory, and for~\\ref{E:graded-monad:kleisli}.,\n \\iffull\\\/see~\\hyperref[S:d1]{Appendix~\\ref{S:d1}}.\\else\\\/see~\\cite{DorschEA19}.\\fi\n\\end{example}\n\n\\noindent One may use the equivalent property of \\autoref{P:d1} to\ndefine depth-1 graded monads over arbitrary base\ncategories~\\cite{MiliusEA15}. We show next that depth-1 graded monads\nmay be specified by giving only $M_0$, $M_1$, the unit~$\\eta$, and $\\mu^{nk}$\nfor $n+k \\leq 1$.\n\n\\begin{theorem}\n\\label{thm:depth-1-graded-monads-M0}\nDepth-$1$ graded monads are in bijective correspondence with\n$6$-tuples $(M_0,M_1,\\ensuremath{\\eta}\\xspace,\\ensuremath{\\mu}\\xspace^{00},\\ensuremath{\\mu}\\xspace^{10},\\ensuremath{\\mu}\\xspace^{01})$ such\nthat the given data satisfy all applicable instances of the graded monad\nlaws.\n\\end{theorem}\n\n\\myparagraph{Semantics via Graded Monads} We next recall how graded\nmonads define \\emph{graded semantics}:\n\\begin{defn}[Graded\n semantics~\\cite{MiliusEA15}]\\label{def:alpha-trace-semantics}\n Given a set functor~$G$, a \\emph{graded semantics} for\n $G$-coalgebras consists of a graded monad\n $((M_n),\\ensuremath{\\eta}\\xspace,(\\ensuremath{\\mu}\\xspace^{nk}))$ and a natural transformation\n $\\alpha\\colon G\\to M_1$. The $\\alpha$-\\emph{pretrace sequence}\n $( \\gamma^{(n)}\\colon X\\to M_nX )_{n<\\omega}$ for a\n $G$-coalgebra $\\gamma\\colon X\\to GX$ is defined by\n \\[\n \\gamma^{(0)} = (X \\xrightarrow{\\ensuremath{\\eta}\\xspace_X} M_0 X)\n \\quad\\text{and}\\quad\n \\gamma^{(n+1)} = (X\n \\xrightarrow{\\alpha_X\\cdot\\gamma} M_1 X \\xrightarrow{M_1\\gamma^{(n)}} M_1 M_n X\n \\xrightarrow{\\ensuremath{\\mu}\\xspace_X^{1n}} M_{n+1}X).\n \\]\n The $\\alpha$-\\emph{trace sequence} $T^\\alpha_\\gamma$ is the sequence\n $( M_n!\\cdot\\gamma^{(n)}\\colon X\\to M_n1)_{n<\\omega}$.\n \n In \\Set, two states $x\\in X$, $y\\in Y$ of coalgebras\n $\\gamma\\colon X\\to GX$ and $\\delta\\colon Y\\to GY$ are $\\alpha$-\\emph{trace} (or\n \\emph{graded}) \\emph{equivalent} if\n $M_n!\\cdot\\gamma^{(n)}(x) = M_n!\\cdot\\delta^{(n)}(y)$ for all\n $n<\\omega$.\n\\end{defn}\nIntuitively, $M_nX$ consists of all length-$n$ \\emph{pretraces}, i.e.\\\ntraces paired with a poststate, and $M_n1$ consists of all length-$n$\ntraces, obtained by erasing the poststate. Thus, a graded semantics\nextracts length-$1$ pretraces from successor structures. In the\nfollowing two examples we have $M_1 = G$; however, in general $M_1$\nand $G$ can differ (\\autoref{sec:ltbt}).\n\n\\begin{example}\\label{expl:monads}\n Recall from \\autoref{sec:prelim} that a $G$-coalgebra for the\n functor $G = \\Pow_\\omega(\\mathcal{A} \\times -)$ is just a finitely branching LTS.\n We recall two graded semantics that model the extreme ends of the\n linear time -- branching time spectrum~\\cite{MiliusEA15}; more\n examples will be given in the next section\n\n \\begin{longitemslist\n \\item \\emph{Trace equivalence.} For $x,y \\in X$ and $w\\in \\mathcal{A}^*$,\n we write $x \\xrightarrow{w} y$ if $y$ can be reached from $x$ on a\n path whose labels yield the word $w$, and\n $\\ensuremath{\\mathcal T}\\xspace(x) = \\{w \\in \\mathcal{A}^* \\mid \\exists y \\in X .\\ x \\xrightarrow{w}\n y\\}$\n denotes the set of \\emph{traces} of $x \\in X$. States $x,y$ are\n \\emph{trace equivalent} if $\\ensuremath{\\mathcal T}\\xspace(x)=\\ensuremath{\\mathcal T}\\xspace(y)$. To capture trace\n semantics of labelled transition systems we consider the graded\n monad with $M_nX = \\mathcal{P}(\\mathcal{A}^n \\times X)$\n \n \n (see \\autoref{E:graded-monad}.\\ref{E:graded-monad:tr}). The\n natural transformation $\\alpha$ is the identity. For a\n $G$-coalgebra $(X, \\gamma)$ and $x \\in X$ we have that\n $\\gamma^{(n)}(x)$ is the set of pairs $( w, y)$ with\n $w \\in \\mathcal{A}^n$ and $x \\xrightarrow{w} y$, i.e.\\ pairs of\n length-$n$ traces and their corresponding poststate. Consequently,\n the~$n$-th component $M_n! \\cdot \\gamma^{(n)}$ of the\n $\\alpha$-trace sequence maps $x$ to the set of its length-$n$\n traces. Thus, $\\alpha$-trace equivalence is standard trace\n equivalence~\\cite{vanglabbeek2001linear}.\n\n Note that the equations presenting the graded monad $M_n$ in\n \\autoref{E:graded-monad}.\\ref{E:graded-monad:tr} bear a striking\n resemblance to the ones given by van Glabbeek to axiomatize\n trace equivalence of processes, with the difference that in his\n axiomatization actions do not distribute over the empty join. In\n fact, $a.0 = 0$ is clearly not valid for processes under trace\n equivalence. In the graded setting, this equation just expresses\n the fact that a trace which ends in a deadlock after $n$ steps\n cannot be extended to a trace of length $n+1$.\n\n \\item\\label{expl:monads:2} \\emph{Bisimilarity.} By the discussion\n of the final sequence of a functor~$G$ (\\autoref{sec:prelim}),\n the graded monad with $M_nX = G^nX$\n (\\autoref{E:graded-monad}.\\ref{E:graded-monad:bisim}), with\n $\\alpha$ being the identity again, captures finite-depth\n behavioural equivalence, and hence behavioural equivalence\n when~$G$ is finitary. In particular, on finitely branching LTS,\n $\\alpha$-trace equivalence is bisimilarity in this case.\n \\end{longitemslist}\n\\end{example}\n\n\\section{A Spectrum of Graded Monads}\\label{sec:ltbt}\n\nWe present graded monads for a range of equivalences on the linear\ntime -- branching time spectrum as well as probabilistic trace\nequivalence for generative probabilistic systems (GPS), giving in each\ncase a graded theory and a description of the arising graded\nmonads. Some of our equations bear some similarity to van Glabbeek's\naxioms for equality of process terms. There are also important\ndifferences, however. In particular, some of van Glabbeek's axioms\nare implications, while ours are purely equational; moreover, van\nGlabbeek's axioms sometimes nest actions, while we employ only\ndepth-$1$ equations (which precludes nesting of actions) in order to\nenable the extraction of characteristic logics later. All graded\ntheories we introduce contain the theory of join semilattices, or in\nthe case of GPS convex algebras, whose operations are assigned\ndepth~$0$; we mention only the additional operations needed. We use\nterminology introduced in \\autoref{expl:monads}.\n\n\\myparagraph{Completed Trace\n Semantics}\\label{ssec:completed-trace-semntics} refines trace\nsemantics by distinguishing whether traces can end in a deadlock. We\ndefine a depth-$1$ graded theory by extending the graded theory for\ntrace semantics (\\autoref{E:graded-monad}) with a constant depth-$1$\noperation~$\\star$ denoting deadlock. The induced graded monad has\n$M_0 X= \\Pow_\\omega(X)$, $M_1 = \\Pow_\\omega(\\mathcal{A} \\times X + 1)$ (and\n$M_nX=\\Pow_\\omega(\\mathcal{A}^n\\times X+\\mathcal{A}^{ 0$\n(i.e.\\ $p+q>0$) in the last equation~\\cite{Jacobs10}. Again, we have\ndepth-$1$ operations $\\sigma$ for action $\\sigma\\in\\mathcal{A}$, now\nsatisfying the equations\n\\begin{math}\n \\sigma(x \\boxplus_p y) = \\sigma(x) \\boxplus_p \\sigma(y).\n\\end{math}\n\n\n\n\n\n\\section{Graded Logics}\\label{sec:logics}\n\n\n\n\\noindent Our next goal is to extract \\emph{characteristic logics}\nfrom graded monads in a systematic way, with \\emph{characterizing}\nmeaning that states are logically indistinguishable iff they are\nequivalent under the semantics at hand. We will refer to these logics\nas \\emph{graded logics}; the implication from graded equivalence to\nlogical indistinguishability is called \\emph{invariance}, and the\nconverse implication \\emph{expressiveness}. E.g.\\ standard modal logic\nwith the full set of Boolean connectives is invariant under\nbisimilarity, and the corresponding expressiveness result is known as\nthe \\emph{Hennessy-Milner theorem}. This result has been lifted to\ncoalgebraic generality early on, giving rise to the \\emph{coalgebraic\n Hennessy-Milner theorem}~\\cite{Pattinson04,Schroder08}. In previous\nwork~\\cite{MiliusEA15}, we have related graded semantics to modal\nlogics extracted from the graded monad in the envisaged fashion. These\nlogics are invariant by construction; the main new result we\npresent here is a generic \\emph{expressiveness} criterion, to be\ndiscussed in \\autoref{sec:expr}. The key ingredient in this criterion\nare \\emph{canonical} graded algebras, which we newly introduce here,\nproviding a recursive-evaluation style reformulation of the semantics\nof graded logics.\n\nA further key issue in characteristic modal logics is the choice of\npropositional operators; e.g.\\ notice that when $\\trdiamond{\\sigma}$\ndenotes the usual Hennessy-Milner style diamond operator for an\naction~$\\sigma$, the formula\n$\\trdiamond{\\sigma}\\top\\land\\trdiamond{\\tau}\\top$ is invariant under\ntrace equivalence (i.e.~the corresponding property is closed under\nunder trace equivalence) but the formula\n$\\trdiamond{\\sigma}(\\trdiamond{\\sigma}\\top\\land\\trdiamond{\\tau}\\top)$,\nbuilt from the former by simply prefixing with~$\\trdiamond{\\sigma}$,\nis not, the problem being precisely the use of conjunction. While in\nour original setup, propositional operators were kept implicit, that\nis, incorporated into the set of modalities, we provide an explicit\ntreatment of propositional operators in the present paper. Besides\nadding transparency to the syntax and semantics, having first-class\npropositional operators will be a prerequisite for the formulation of\nthe expressiveness theorem.\n\n\\myparagraph{Coalgebraic Modal Logic} To provide context, we briefly\nrecall the setup of \\emph{coalgebraic modal\n logic}~\\cite{Pattinson04,Schroder08}. Let~$2$ denote the\nset~$\\{\\bot,\\top\\}$ of Boolean truth values; we think of the set~$2^X$\nof maps $X\\to 2$ as the set of predicates on~$X$. Coalgebraic logic in\ngeneral abstracts systems as coalgebras for a functor~$G$, like we do\nhere; fixes a set~$\\Lambda$ of \\emph{modalities} (unary for the sake\nof readability); and then interprets a modality $L\\in\\Lambda$ by the\nchoice of a \\emph{predicate lifting}, i.e.\\ a natural transformation\n\\begin{equation*}\n \\Sem{L}_X\\colon 2^X\\to 2^{GX}.\n\\end{equation*}\nBy the Yoneda lemma, such natural transformations are in bijective\ncorrespondence with maps $G2\\to 2$~\\cite{Schroder08}, which we shall\nalso denote as $\\Sem{L}$. In the latter formulation, the recursive\nclause defining the interpretation $\\Sem{L\\phi}\\colon X\\to 2$, for a\nmodal formula~$\\phi$, as a state predicate in a $G$-coalgebra\n$\\gamma \\colon X\\to GX$ is then\n\\begin{equation}\\label{eq:coalg-modality}\n \\Sem{L\\phi}= (X\\xrightarrow{\\gamma}GX\\xrightarrow{ G\\Sem{\\phi}} G2\\xrightarrow{\\Sem{L}}2).\n\\end{equation}\nE.g.\\ taking $G=\\Pow_\\omega(\\mathcal{A}\\times-)$ (for labelled transition systems),\nwe obtain the standard semantics of the Hennessy-Milner diamond\nmodality $\\trdiamond{\\sigma}$ for~$\\sigma\\in\\mathcal{A}$ via the predicate\nlifting\n\\begin{equation*}\n \\Sem{\\trdiamond{\\sigma}}_X(f)=\\{B\\in\\Pow_\\omega(\\mathcal{A}\\times X)\\mid\n \\exists x.\\,(\\sigma,x)\\in B\\land f(x)=\\top\\}\\qquad(\\text{for\n $f\\colon X\\to 2$}).\n\\end{equation*}\nIt is easy to see that \\emph{coalgebraic modal logic}, which combines\ncoalgebraic modalities with the full set of Boolean connectives, is\ninvariant under finite-depth behavioural equivalence\n(\\autoref{sec:prelim}). Generalizing the classical Hennessy-Milner\ntheorem~\\cite{HennessyMilner85}, the \\emph{coalgebraic Hennessy-Milner\n theorem}~\\cite{Pattinson04,Schroder08} shows that conversely,\ncoalgebraic modal logic \\emph{characterizes} behavioural equivalence,\ni.e.\\ logical indistinguishability implies behavioural equivalence,\nprovided that~$G$ is finitary (implying coincidence of behavioural\nequivalence and finite-depth behavioural equivalence) and~$\\Lambda$ is\n\\emph{separating}, i.e.\\ for every finite set~$X$, the set\n\\begin{equation*}\n \\Lambda(2^X)=\\{\\Sem{L}(f)\\mid f\\in 2^X\\}\n\\end{equation*}\nof maps $GX\\to 2$ is jointly injective.\n\nWe proceed to introduce the syntax and semantics of graded logics.\n\\myparagraph{Syntax} We parametrize the syntax of \\emph{graded logics}\nover\n\\begin{itemize}\n\\item a set~$\\Theta$ of \\emph{truth constants},\n\\item a set~$\\mathcal{O}$ of \\emph{propositional operators} with assigned\n finite arities, and\n\\item a set~$\\Lambda$ of \\emph{modalities} with assigned arities.\n\\end{itemize}\nFor readability, we will restrict the technical exposition to unary\nmodalities; the treatment of higher arities requires no more than\nadditional indexing (and we will use $0$-ary modalities in the\nexamples). E.g.\\ standard Hennessy-Milner logic is given by\n$\\Lambda=\\{\\trdiamond{\\sigma}\\mid \\sigma\\in\\mathcal{A}\\}$ and~$\\mathcal{O}$\ncontaining all Boolean connectives. Other logics will be determined by\nadditional or different modalities, and often by fewer propositional\noperators. Formulae of the logic are restricted to have uniform depth,\nwhere propositional operators have depth~$0$ and modalities have\ndepth~$1$; a somewhat particular feature is that truth constants can\nhave top-level occurrences only in depth-$0$ formulae. That is,\nformulae~$\\phi,\\phi_1,\\dots$ of depth~$0$ are given by the grammar\n\\begin{equation*}\n \\phi\\Coloneqq p(\\phi_1,\\dots,\\phi_k) \\mid c\n \\qquad (p\\in\\mathcal{O}\\text{ $k$-ary}, c\\in\\Theta),\n\\end{equation*}\nand formulae~$\\phi$ of depth $n+1$ by\n\\begin{equation*}\n \\phi\\Coloneqq p(\\phi_1,\\dots,\\phi_k) \\mid L\\psi\n \\qquad (p\\in\\mathcal{O}\\text{ $k$-ary}, L\\in\\Lambda)\n\\end{equation*}\nwhere $\\phi_1,\\dots,\\phi_n$ range over formulae of depth $n+1$ and\n$\\psi$ over formulae of depth~$n$. \n\n\\myparagraph{Semantics} The semantics of graded logics is parametrized\nover the choice of \\emph{a functor~$G$, a depth-$1$ graded monad\n $M=((M_n)_{n<\\omega},\\eta,$ $(\\mu^{nk})_{n,k<\\omega})$, and a\n graded semantics~$\\alpha\\colon G\\to M_1$, which we fix for the\n remainder of the paper}. It was originally given by translating\nformulae into \\emph{graded algebras} and then defining formula\nevaluation by the universal property of $(M_n1)$ as a free graded\nalgebra~\\cite{MiliusEA15}; here, we reformulate the semantics in a\nmore standard style by recursive clauses, using canonical graded\nalgebras. In general, the notion of graded algebra is defined as\nfollows~\\cite{MiliusEA15}.\n\\begin{defn}[Graded algebras]\n Let $n<\\omega$. A \\emph{(graded) $M_n$-algebra}\n $A=((A_k)_{k\\le n},(a^{mk})_{m+k\\le n})$ consists of carrier\n sets~$A_k$ and structure maps\n \\begin{equation*}\n a^{mk}\\colon M_mA_k\\to A_{m+k}\n \\end{equation*}\n satisfying the laws \n \\begin{equation}\\label{diag:alg}\n \\begin{tikzcd}\n A_k \\arrow{r}{\\eta_{A_k}} \\arrow[equal]{dr\n & M_0 A_k\\arrow{d}{a^{0k}} & \n M_m M_r A_k \\arrow{r}{M_m a^{rk}}\n \\arrow{d}[left]{\\mu^{mr}_{A_k}} &\n M_m A_{r+k} \\arrow{d}{a^{m,r+k}} \\\\\n & A_k & M_{m+r}A_k \\arrow{r}{a^{m+r,k}} & A_{m+r+k}\n \\end{tikzcd}\n \\end{equation}\n for all $k\\le n$ (left) and all $m,r,k$ such that $m+r+k\\le n$\n (right), respectively. An \\emph{$M_n$-morphism}~$f$ from~$A$ to an\n $M_n$-algebra $B=((B_k)_{k\\le n},(b^{mk})_{m+k\\le n})$ consists of\n maps $f_k\\colon A_k\\to B_k$, $k\\le n$, such that\n $f_{m+k}\\cdot a^{mk}=b^{mk}\\cdot M_mf_k$\n \n \n \n \n \n \n \n \n for all $m,k$ such that $m+k\\le n$.\n\\end{defn}\n\\noindent \nWe view the carrier~$A_k$ of an~$M_n$-algebra as the set of algebra\nelements that have already absorbed operations up to depth~$k$. As in\nthe case of plain monads, we can equivalently describe graded algebras\nin terms of graded theories: If $M$ is generated by a graded theory\n$\\mathbb{T}=(\\Sigma,E,d)$, then an $M_n$-algebra interprets each operation\n$f\\in\\Sigma$ of arity~$r$ and depth~$d(f)=m$ by maps\n$f^A_k\\colon A_k^r\\to A_{m+k}$ for all $k$ such that $m+k\\le n$; this\ngives rise to an inductively defined interpretation of terms\n(specifically, given a valuation of variables in~$A_m$, terms of\nuniform depth~$k$ receive values in~$A_{k+m}$, for $k+m\\le n$), and\nsubsequently to the expected notion of satisfaction of equations.\n\n\nWhile in general, graded algebras are monolithic objects, for\n\\mbox{depth-$1$} graded monads we can construct them in a modular\nfashion from $M_1$-algebras~\\cite{MiliusEA15}; we thus restrict\nattention to $M_0$- and $M_1$-algebras in the following. We note that\nan $M_0$-algebra is just an Eilenberg-Moore algebra for the\nmonad~$M_0$. An $M_1$-Algebra~$A$ consists of $M_0$-algebras\n$(A_0,a^{00}\\colon M_0A_0\\to A_0)$ and $(A_1,a^{01}\\colon M_0A_1\\to A_1)$, and a\n\\emph{main structure map} $a^{10}\\colon M_1A_0\\to A_1$ satisfying two\ninstances of the right-hand diagram in~\\eqref{diag:alg}, one of which\nsays that $a^{10}$ is a morphism of $M_0$-algebras\n(\\emph{homomorphy}), and the other that the diagram\n\\begin{equation}\n \\label{diag:algebra-coeq}\n \\begin{tikzcd}[column sep=large]\n M_1M_0A_0 \\arrow[shift left]{r}[above]{\\mu^{10}}\n \\arrow[shift right]{r}[below]{M_1a^{00}}& M_1A_0 \\arrow{r}{a^{10}} & A_1,\n \\end{tikzcd}\n\\end{equation}\nwhich by the laws of graded monads consists of $M_0$-algebra\nmorphisms, commutes (\\emph{coequalization}). We will often refer to an\n$M_1$-algebra by just its main structure map.\n\nWe will use $M_1$-algebras as interpretations of the modalities in\ngraded logics, generalizing the previously recalled interpretation of\nmodalities as maps $G2\\to 2$ in branching-time coalgebraic modal\nlogic. We fix an $M_0$-algebra $\\Omega$ of \\emph{truth values}, with\nstructure map $o\\colon M_0\\Omega\\to\\Omega$ (e.g.\\\nfor~$G=\\Pow_\\omega$, $\\Omega$ is a join semilattice). Powers~$\\Omega^n$\nof~$\\Omega$ are again\n$M_0$-algebras.\nA modality $L\\in\\Lambda$ is interpreted as an $M_1$-algebra\n$A=\\Sem{L}$ with carriers $A_0=A_1=\\Omega$ and\n$a^{01}=a^{00}=o$. Such an $M_1$-algebra is thus specified\nby its main structure map $a^{10}\\colon M_1\\Omega\\to\\Omega$ alone, so\nfollowing the convention indicated above we often write $\\Sem{L}$ for\njust this map.\nThe evaluation of modalities is defined using canonical\n$M_1$-algebras:\n\\begin{defn}[Canonical algebras]\n The \\emph{$0$-part} of an~$M_1$-algebra~$A$ is the $M_0$-algebra\n $(A_0,a^{00})$. Taking $0$-parts defines a functor $U_0$ from\n $M_1$-algebras to $M_0$-algebras. An $M_1$-algebra is\n \\emph{canonical} if it is free, w.r.t.\\ $U_0$, over its\n $0$-part. For~$A$ canonical and a modality $L\\in\\Lambda$, we\n denote the unique morphism $A_1\\to\\Omega$ extending an\n $M_0$-morphism $f\\colon A_0\\to\\Omega$ to an $M_1$-morphism $A\\to\\Sem{L}$\n by~$\\Sem{L}(f)$, i.e.\\ $\\Sem{L}(f)$ is the unique $M_0$-morphism\n such that\n \\iffull\n the square below commutes:\n \\begin{equation}\\label{diag:L(f)}\n \\begin{tikzcd}\n M_1 A_0 \\arrow{r}{M_1f} \\arrow{d}[left]{a^{10}} \n & M_1\\Omega\\arrow{d}{\\Sem{L}}\\\\\n A_1 \\arrow{r}[below]{\\Sem{L}(f)} & \\Omega\n \\end{tikzcd}\n \\end{equation}\n \\else\n the following equation holds:\n \\begin{equation}\\label{diag:L(f)}\n (M_1 A_0 \\xrightarrow{M_1 f} M_1\\Omega \\xrightarrow{\\Sem{L}} \\Omega)\n =\n (M_1 A_0 \\xrightarrow{a^{10}} A_1 \\xrightarrow{\\Sem{L}(f)} \\Omega).\n \\end{equation}\n \\fi\n\\end{defn}\n\\begin{lemma}\\label{lem:canonical}\n An $M_1$-algebra~$A$ is canonical iff \\eqref{diag:algebra-coeq} is a\n (reflexive) coequalizer diagram in the category of $M_0$-algebras.\n\\end{lemma}\n\\noindent By the above lemma, we obtain a key example of canonical\n$M_1$-algebras:\n\\begin{corollary}\n If $M$ is a depth-$1$ graded monad, then for every~$n$ and every\n set~$X$, the $M_1$-algebra with carriers $M_nX,M_{n+1}X$ and\n multiplication as algebra structure is canonical.\n\\end{corollary}\n \n\n\n\\noindent Further, we interpret truth constants $c\\in\\Theta$ as\nelements of~$\\Omega$, understood as maps $\\hat{c}\\colon 1\\to\\Omega$,\nand $k$-ary propositional operators $p\\in\\mathcal{O}$ as $M_0$-homomorphisms\n$\n\\Sem{p}\\colon\\Omega^k\\to\\Omega.\n$\nIn our examples on the linear time -- branching time spectrum,~$M_0$\nis either the identity or, most of the time, the finite powerset\nmonad. In the former case, all truth functions are $M_0$-morphisms. In\nthe latter case, the $M_0$-morphisms $\\Omega^k\\to \\Omega$ are the\njoin-continuous functions; in the standard case where $\\Omega=2$ is\nthe set of Boolean truth values, such functions~$f$ have the form\n$f(x_1,\\dots,x_k)=x_{i_1}\\lor\\dots\\lor x_{i_l}$, where\n$i_1,\\dots,i_l\\in\\{1,\\dots,k\\}$. We will see one case where $M_0$ is\nthe distribution monad; then $M_0$-morphisms are affine\nmaps.\n\nThe semantics of a formula~$\\phi$ in graded logic is defined recursively\nas an $M_0$-morphism\n$\\Sem{\\phi}\\colon (M_n1, \\mu^{0n}_1) \\to (\\Omega, o)$ by\n\\begin{equation*}\n \\Sem{c} = (M_01\\xrightarrow{M_0\\hat c}M_0\\Omega \\xrightarrow{o}\\Omega)\\quad\n \\Sem{p(\\phi_1,\\dots,\\phi_k)} =\\Sem{p}\\cdot\\langle\\Sem{\\phi_1},\\dots,\n \\Sem{\\phi_k}\\rangle\\quad\n \\Sem{L\\phi} = \\Sem{L}(\\Sem{\\phi}).\n\\end{equation*}\nThe evaluation of~$\\phi$ in a coalgebra $\\gamma\\colon X\\to GX$ is then given\nby composing with the trace sequence, i.e.\\ as\n\\begin{equation}\\label{eq:formula-eval}\n X\\xrightarrow{M_n!\\cdot\\gamma^{(n)}} M_n1\\xrightarrow{\\Sem\\phi}\\Omega.\n\\end{equation}\nIn particular, graded logics are, by construction, invariant under the\ngraded semantics. \n\n \n\n\\begin{example}[Graded logics]\\label{expl:logics}\n We recall the two most basic examples, fixing $\\Omega=2$ in both\n cases, and $\\top$ as the only truth constant:\n \\begin{longitemslist\n \\item \\emph{Finite-depth behavioural equivalence:} Recall that the\n graded monad $M_nX=G^nX$ captures finite-depth behavioural\n equivalence on $G$-coalgebras. Since~$M_0$ is the identity monad,\n $M_0$-algebras are just sets. Thus, every function $2^k\\to 2$ is\n an $M_0$-morphism,\n \n so we can use all Boolean operators as propositional\n operators. Moreover, $M_1$-algebras are just maps\n $a^{10}\\colon GA_0\\to A_1$. Such an $M_1$-algebra is canonical iff\n $a^{10}$ is an isomorphism, and modalities are interpreted as\n $M_1$-algebras $G2\\to 2$, with the evaluation according\n to~\\eqref{diag:L(f)} and~\\eqref{eq:formula-eval} corresponding\n precisely to the semantics of modalities in coalgebraic\n logic~\\eqref{eq:coalg-modality}. Summing up, we obtain precisely\n coalgebraic modal logic as summarized above in this case. In our\n running example $G=\\Pow_\\omega(\\mathcal{A}\\times(-))$, we take modalities\n $\\Diamond_\\sigma$ as above, with\n $\\Sem{\\Diamond_\\sigma}\\colon\\Pow_\\omega(\\mathcal{A}\\times 2)\\to 2$ defined by\n $\\Sem{\\Diamond_\\sigma}(S)=\\top$ iff $(\\sigma,\\top)\\in S$,\n obtaining precisely classical Hennessy-Milner\n logic~\\cite{HennessyMilner85}.\n \\item \\emph{Trace equivalence:} Recall that the trace semantics of\n labelled transition systems with actions in~$\\mathcal{A}$ is modelled by\n the graded monad $M_nX=\\Pow_\\omega(\\mathcal{A}^n\\times X)$. As indicated above,\n in this case we can use disjunction as a propositional operator\n since $M_0=\\Pow_\\omega$. Since the graded theory for $M_n$ specifies for\n each $\\sigma\\in\\mathcal{A}$ a unary depth-$1$ operation that distributes\n over joins, we find that the maps $\\Sem{\\Diamond_\\sigma}$ from the\n previous example (unlike their duals $\\Box_\\sigma$) induce\n $M_1$-algebras also in this case, so we obtain a graded trace\n logic featuring precisely diamonds and disjunction, as expected.\n \\end{longitemslist}\n We defer the discussion of further examples, including ones where\n $\\Omega=[0,1]$, to the next section, where we will simultaneously\n illustrate the generic expressiveness result\n (\\autoref{expl:ltbt-logics}). \n\\end{example}\n\n\\begin{remark}\n One important class of examples where the above approach to\n characteristic logics will \\emph{not} work without substantial\n further development are simulation-like equivalences, whose\n characteristic logics need\n conjunction~\\cite{vanglabbeek2001linear}. Conjunction is not an\n $M_0$-morphism for the corresponding graded monads identified in\n \\autoref{sec:ltbt}, which both have $M_0=\\Pow_\\omega$. A related and maybe\n more fundamental observation is that formula evaluation is not\n $M_0$-morphic in the presence of conjunction; e.g.\\ over simulation\n equivalence, the evaluation map\n $M_11=\\Pow_\\omega^\\downarrow(\\mathcal{A}\\times\\Pow_\\omega(1))\\to 2$ of the formula\n $\\trdiamond{\\sigma}\\top\\land\\trdiamond{\\tau}\\top$ fails to be\n join-continuous for distinct $\\sigma,\\tau\\in\\mathcal{A}$. We leave the\n extension of our logical framework to such cases to future work,\n expecting a solution in elaborating the theory of graded monads,\n theories, and algebras over the category of partially ordered sets,\n where simulations live more naturally (e.g.~\\cite{KapulkinEA12}).\n\\end{remark}\n\n\n\n\n\n\n\n\\section{Expressiveness}\\label{sec:expr}\n\nWe now present our main result, an expressiveness criterion for graded\nlogics, which states that a graded logic characterizes the given\ngraded semantics if it has enough modalities propositional operators,\nand truth constants. Both the criterion and its proof now fall into\nplace naturally and easily, owing to the groundwork laid in the\nprevious section, in particular the reformulation of the semantics in\nterms of canonical algebras:\n\\begin{defn}\\label{def:separation}\n We say that a graded logic with set~$\\Omega$ of truth values and\n sets~$\\Theta$,~$\\mathcal{O}$,~$\\Lambda$ of truth constants,\n propositional operators, and modalities, respectively, is\n \\begin{longitemslist}\n \\item \\emph{depth-$0$ separating} if the family of maps\n $\\Sem{c}\\colon M_01\\to\\Omega$, for truth constants\n $c\\in\\Theta$, is jointly injective; and\n \\item \\emph{depth-$1$ separating} if, whenever $A$ is a canonical\n $M_1$-algebra and $\\mathfrak{A}$ is a jointly injective set of\n $M_0$-homomorphisms $A_0\\to\\Omega$ that is closed under the\n propositional operators in~$\\mathcal{O}$ (in the sense that\n $\\Sem{p}\\cdot \\langle f_1,\\dots,f_k\\rangle\\in\\mathfrak{A}$ for\n $f_1,\\dots,f_k\\in\\mathfrak{A}$ and $k$-ary $p\\in\\mathcal{O}$), then the\n set \n \\[\n \\Lambda(\\mathfrak{A})\\coloneqq\\{\\Sem{L}(f)\\colon A_1\\to\\Omega\\mid L\\in\\Lambda,f\\in\\mathfrak{A}\\}.\n \\]\n of maps is jointly injective.\n \\end{longitemslist}\n\\end{defn}\n\\begin{theorem}[Expressiveness]\\label{thm:expr}\n If a graded logic is both depth-$0$ separating and depth-$1$\n separating, then it is expressive.\n\\end{theorem}\n\n\\begin{example}[Logics for bisimilarity]\\label{expl:bisim-logics}\n We note first that the existing coalgebraic Hennessy-Milner theorem,\n for branching time equivalences and coalgebraic modal logic with\n full Boolean base over a finitary\n functor~$G$~\\cite{Pattinson04,Schroder08}, as recalled in\n Section~\\ref{sec:logics}, is a special case of \\autoref{thm:expr}:\n We have already seen in \\autoref{expl:logics} that coalgebraic modal\n logic in the above sense is an instance of our framework for the\n graded monad $M_nX=G^nX$. Since $M_0=\\id$ in this case, depth-$0$\n separation is vacuous. As indicated in \\autoref{expl:logics},\n canonical $M_1$-algebras are w.l.o.g.\\ of the form $\\id\\colon GX\\to GX$,\n where for purposes of proving depth-$1$ separation, we can restrict\n to finite~$X$ since~$G$ is finitary. Then, a set~$\\mathfrak{A}$ as in\n \\autoref{def:separation} is already the whole powerset $2^X$, so\n depth-$1$ separation is exactly the previous notion of separation.\n \n A well-known particular case is probabilistic bisimilarity on Markov\n chains, for which an expressive logic needs only probabilistic\n modalities $\\Diamond_p$ `with probability at least~$p$' and\n conjunction~\\cite{DesharnaisEA98}. This result (later extended to\n more complex composite functors~\\cite{MossViglizzo06}) is also\n easily recovered as an instance of \\autoref{thm:expr}, using the\n same standard lemma from measure theory as in \\emph{op.~cit.},\n which states that measures are uniquely determined by their values\n on a generating set of the underlying $\\sigma$-algebra that is\n closed under finite intersections (corresponding to the set~$\\mathfrak{A}$\n from \\autoref{def:separation} being closed under conjunction).\n\\end{example}\n\n\\begin{remark}\n For behavioural equivalence, i.e.\\ $M_nX=G^nX$ as in the above\n example, the inductive proof of our expressiveness theorem\n essentially instantiates to Pattinson's proof of the coalgebraic\n Hennessy-Milner theorem by induction over the terminal\n sequence~\\cite{Pattinson04}. One should note that although the\n coalgebraic Hennessy-Milner theorem can be shown to hold for larger\n cardinal bounds on the branching by means of a direct quotienting\n construction~\\cite{Schroder08}, the terminal sequence argument goes\n beyond finite branching only in corner cases.\n\\end{remark}\n\n\\begin{example}[Expressive graded logics on the linear time -- branching time spectrum]\\label{expl:ltbt-logics}\n We next extract graded logics from some of the graded monads\n for the linear time -- branching time spectrum introduced in\n \\autoref{sec:ltbt}, and show how in each case, expressiveness is an\n instance of \\autoref{thm:expr}. Bisimilarity is already covered by\n the previous example. Depth-$0$ separation is almost always trivial\n and not mentioned further. Unless mentioned otherwise, all logics\n have disjunction, enabled by $M_0$ being powerset as discussed in\n the previous section. Most of the time, the logics are essentially\n already given by van Glabbeek (with the exception that we show that\n one can add disjunction)~\\cite{vanglabbeek2001linear}; the emphasis\n is entirely on uniformization.\n \\begin{longitemslist\n \\item \\emph{Trace equivalence:} As seen in \\autoref{expl:logics},\n the graded logic for trace equivalence features (disjunction and)\n diamond modalities $\\Diamond_\\sigma$ indexed over actions\n $\\sigma\\in\\mathcal{A}$. The ensuing proof of depth-$1$ separation uses\n canonicity of a given $M_1$-algebra~$A$ only to obtain that the\n structure map $a^{10}$ is surjective. The other key point is that\n a jointly injective collection~$\\mathfrak{A}$ of $M_0$-homomorphisms\n $A_0\\to 2$, i.e.\\ join preserving maps, has the stronger\n separation property that whenever $x\\not\\le y$ then there exists\n $f\\in\\mathfrak{A}$ such that $f(x)=\\top$ and $f(y)=\\bot$.\n \\item Graded logics for completed traces, readiness, failures, ready\n traces, and failure traces are developed from the above by adding\n constants or additionally indexing modalities over sets of\n actions, with only little change to the proofs of depth-$1$\n separation. For completed trace equivalence, we just add a $0$-ary\n modality $\\star$ indicating deadlock. For ready trace equivalence,\n we index the diamond modalities $\\Diamond_\\sigma$ with sets\n $I\\subseteq\\mathcal{A}$; formulae $\\Diamond_{\\sigma,I}\\phi$ are then read\n `the current ready set is~$I$, and there is a $\\sigma$-successor\n satisfying~$\\phi$'. For failure trace equivalence we proceed in\n the same way but read the index~$I$ as `$I$ is a failure set at\n the current state'. For readiness equivalence and failures\n equivalence, we keep the modalities~$\\Diamond_\\sigma$ unchanged\n from trace equivalence and instead introduce $0$-ary\n modalities~$r_I$ indicating that~$I$ is the ready set or a failure\n set, respectively, at the current state, thus ensuring that\n formulae do not continue after postulating a ready set.\n \\end{longitemslist}\n\\end{example}\n\n\\begin{example}[Probabilistic traces]\\label{expl:prob-trace}\n We have recalled in \\autoref{sec:ltbt} that probabilistic trace\n equivalence of generative probabilistic transition systems can be\n captured as a graded semantics using the graded\n monad~$M_nX=\\mathcal{D}(\\mathcal{A}^n\\times X)$, with $M_0$-algebras being convex\n algebras. In earlier work~\\cite{MiliusEA15} we have noted that a\n logic over the set $\\Omega=[0,1]$ of truth values (with the usual\n convex algebra structure) featuring rational truth constants, affine\n combinations as propositional operators (as indicated in\n \\autoref{sec:logics}), and modal operators $\\langle\\sigma\\rangle$,\n interpreted by $M_1$-algebras\n $\\Sem{\\langle\\sigma\\rangle}\\colon M_1[0,1]\\to[0,1]$ defined by\n \\iffull\n \\begin{equation*}\\textstyle\n \\Sem{\\langle\\sigma\\rangle}(\\mu)=\\sum_{r\\in [0,1]}r\\mu(\\sigma,r)\n \\end{equation*}\n \\else\\\/$\\Sem{\\langle\\sigma\\rangle}(\\mu)=\\sum_{r\\in\n [0,1]}r\\mu(\\sigma,r)$ \\fi\n is invariant under probabilistic trace equivalence. By our\n expressiveness criterion, we recover the result that this logic\n is expressive for probabilistic trace semantics\n (see e.g.~\\cite{bernardo-botta:characterising-logics}).\n\\end{example}\n\\section{Conclusion and Future Work}\n\nWe have provided graded monads modelling a range of process\nequivalences on the linear time -- branching time spectrum, presented\nin terms of carefully designed graded algebraic theories. From these\ngraded monads, we have extracted characteristic modal logics for the\nrespective equivalences systematically, following a paradigm of graded\nlogics that grows out of a natural notion of graded algebra. Our main\ntechnical results concern the further development of the general\nframework for graded logics; in particular, we have introduced a\nfirst-class notion of propositional operator, and we have established\na criterion for \\emph{expressiveness} of graded logics that\nsimultaneously takes into account the expressive power of the\nmodalities and that of the propositional base. (An open question that\nremains is whether an expressive logic always exists, as it does in\nthe branching-time setting~\\cite{Schroder08}.) Instances of this\nresult include, for instance, the coalgebraic Hennessy-Milner\ntheorem~\\cite{Pattinson04,Schroder08}, Desharnais et al.'s\nexpressiveness result for probabilistic modal logic with only\nconjunction~\\cite{DesharnaisEA98}, and expressiveness for various\nlogics for trace-like equivalences on non-deterministic and\nprobabilistic systems. The emphasis in the examples has been on\nwell-researched equivalences and logics for the basic case of labelled\ntransition systems, aimed at demonstrating the versatility of graded\nmonads and graded logics along the axis of granularity of system\nequivalence. The framework as a whole is however parametric also over\nthe branching type of systems and in fact over the base category\ndetermining the structure of state spaces; an important direction for\nfuture research is therefore to capture (possibly new) equivalences\nand extract expressive logics on other system types such as\nprobabilistic systems (we have already seen probabilistic trace\nequivalence as an instance; see~\\cite{BonchiEA17} for a comparison of\nsome equivalences on probabilistic automata, which combine\nprobabilities and non-determinism) and nominal systems, e.g.\\ nominal\nautomata~\\cite{BojanczykEA14,SchroderEA17}. Moreover, we plan to\nextend the framework of graded logics to cover also temporal logics,\nusing graded algebras of unbounded depth.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section*{Introduction}\n \nThe purpose of this paper is to prove some saturation bounds for the ideals of non-singular complex projective varieties and their powers.\n\n\n\nWe begin with some background. Consider the polynomial ring $S = \\mathbf{C}[x_0, \\ldots, x_r]$ in $r+1$ variables, and fix homogeneous polynomials\n\\[f_0\\, , \\, f_1 \\, , \\, \\ldots \\, , \\, f_p\\, \\in \\, S \\ \\ \\text{with\\ \\ $\\deg(f_i) = d_i$}.\\] We assume that\n$d_0 \\ge d_1 \\ge \\ldots \\ge d_p$, \nand we denote by \n\\[ J \\ = \\ \\big ( \\, f_0 \\, , \\, f_1 \\, , \\, \\ldots \\, ,\\, f_p \\, \\big) \\ \\subseteq \\ S \\]\nthe ideal that the polynomials span. Suppose now that $J$ is primary for the irrelevant maximal ideal $\\mathfrak{m} = (x_0, \\ldots, x_r)$, or equivalently that $\\dim_\\mathbf{C} S\/J < \\infty$. In this case $J$ contains all monomials of sufficiently large degree, and it is a classical theorem of Macaulay \\cite[Theorem 7.4.1]{CM-S.P} that \n\\begin{equation} \\label{Macaulay.Eqn.1}\nJ_t \\ = \\ S_t \\ \\ \\text{ for } \\ \\ t\\, \\ge \\, d_0 + \\ldots + d_r - r. \n\\end{equation}\nMoreover this bound is (always) sharp when $p = r$. \nAlthough less well known, a similar statement holds for powers of $J$:\n\\begin{equation} \\label{Macaulay.Eqn.2}\n(J^a)_t \\ = \\ S_t \\ \\ \\text{ for } \\ \\ t\\, \\ge \\, ad_0 +d_1 + \\ldots + d_r - r.\\end{equation}\nThis again is always sharp when $ p = r$. \n\nIt is natural to ask whether there are analogous results for more general homogeneous ideals $J$, in particular when \n\\[ X \\ =_{\\text{def}} \\ \\Zero{J} \\ \\subseteq \\ \\mathbf{P}^r \\]\nis a smooth complex projective variety. Of course if $J$ has non-trivial zeroes, then it does not contain any power of the maximal ideal. However if one interprets \\eqref{Macaulay.Eqn.1} and \\eqref{Macaulay.Eqn.2} as saturation bounds, then the question makes sense more generally. Specifically, recall that the \\textit{saturation} of a homogeneous ideal $J$ is defined by\n\\begin{equation} \\satt{J} \\ = \\ \\big \\{ \\, f \\in S \\mid \\mathfrak{m}^k \\cdot f \\subseteq J \\text{ for some $k \\ge 0$} \\, \\big \\}. \\notag \\end{equation}\nThe quotient $ \\satt{J} \/ J$ has finite length, and in particular\n\\[ ( \\satt{J} )_t \\ = \\ J_t \\ \\ \\text{ for } \\ t \\gg 0. \\]\nThe least such integer $t$ is called the \\textit{saturation degree} $\\textnormal{sat. \\!deg}(J)$ of $J$. Observing that $\\satt{J} = S$ if and only if $J$ is $\\mathfrak{m}$-primary, statements \\eqref{Macaulay.Eqn.1} and \\eqref{Macaulay.Eqn.2} are equivalent to estimates for the saturation degrees of $J$ and $J^a$. So the problem becomes to bound the saturation degree of an ideal in terms of the degrees of its generators.\n\nIt is instructive to consider some examples. Let $X \\subseteq \\mathbf{P}^r$ be a hyperplane defined by a linear form $\\ell \\in S$, and set\n\\begin{equation} \\label{Hyperplane.Example} f_i \\, = \\, x_i^{d-1}\\cdot \\ell \\ \\ \\ , \\ \\ \\ J = (f_0, \\ldots, f_r)\\, \\subseteq \\, S. \\end{equation}\nThen $\\satt{J} = (\\ell)$, and it follows from Macaulay's theorem that\n\\[ \\textnormal{sat. \\!deg}(J) \\ = \\ (r+1)(d-1) - r + 1 \\ = \\ (r+1)d - 2r, \\]\nwhich is very close to the bound \\eqref{Macaulay.Eqn.1}. On the other hand, it is not the case that the saturation degree of an arbitrary ideal is bounded linearly in the degrees of its generators. For instance, \n the ideals \\[ J \\ = \\ \\big( x^d, y^d, xz^{d-1} - yw^{d-1} \\big) \\ \\subseteq \\ \\mathbf{C}[x,y,z,w] \\]\nconsidered by Caviglia \\cite[Example 4.2.1]{Caviglia} have $\\textnormal{sat. \\!deg}(J) \\approx d^2$.\n\nOur first main result asserts that for ideals defining smooth varieties, the Macaulay bounds remain true without modification.\n\\begin{theoremalpha} \\label{Intro.Sat.Deg.Thm}\nAs above, suppose that \n\\[ J \\ = \\ \\big ( \\, f_0 \\, , \\, f_1 \\, , \\, \\ldots \\, ,\\, f_p \\, \\big) \\ \\subseteq \\ S \\]\nis generated by forms of degrees $d_0 \\ge \\ldots \\ge d_p$, and assume that the projective scheme\n\\[ X \\ =_{\\text{def}} \\ \\Zero{J} \\ \\subseteq \\ \\mathbf{P}^r \\]\ncut out by the $f_i$ is a non-singular complex variety. Then \n$ \\textnormal{sat. \\!deg}(J) \\le d_0 + \\ldots + d_r - r$, \nand more generally\n\\begin{equation} \\label{Intro.Thm.Equation} \\textnormal{sat. \\!deg}(J^a) \\ \\le \\ ad_0 + d_1 + \\ldots + d_r - r. \\end{equation}\n\\end{theoremalpha}\n\\noindent (If $p < r$, one takes $d_{p+1} = \\ldots = d_r = 0$.) We do not know whether the stated bound is best possible, but in any event it is asymptotically sharp. Indeed, if $J$ is the ideal considered in \\eqref{Hyperplane.Example}, then the Theorem predicts that $\\textnormal{sat. \\!deg}(J^a)\\le (a+r)d -r$, whereas in fact $\\textnormal{sat. \\!deg}(J^a) = (a+r)d - 2r$. \n\n\n\n\nGiven a reduced algebraic set $X \\subseteq \\mathbf{P}^r$ denote by $I_X \\subseteq S$ the saturated homogeneous ideal of $X$. Recall that the \\textit{symbolic powers} of $I_X$ are\n\\[\nI_X^{(a)} \\ = \\ \\big \\{ f \\in S \\mid \\textnormal{ord}_x(f) \\ge a \\text{ for general (or every) } x\\in X \\, \\big \\}. \n\\]\nEvidently $I_X^a \\subseteq I_X^{(a)}$, and there has been a huge amount of interest in recent years in understanding the connections between actual and symbolic powers (cf \\cite{ELS}, \\cite{Hochster.Huneke}, \\cite{BocciHarbourne}, \\cite{Dao.ea}). If $X$ is non-singular, then $I_X^{(a)} = \\satt{(I_X^a)}$. Therefore Theorem \\ref{Intro.Sat.Deg.Thm} implies\n\\begin{corollaryalpha}\nAssume that $X \\subseteq \\mathbf{P}^r$ is smooth, and that $I_X$ is generated in degrees $d_0 \\ge d_1 \\ge \\ldots \\ge d_p$. Then\n\\[ \\big ( I_X^{(a)} \\big)_t \\ = \\ ( I_X^a)_t \\ \\ \\text{ for } \\, t \\, \\ge \\, ad_0 +d_1 + \\ldots + d_r -r. \\]\n\\end{corollaryalpha}\n\\noindent \nFor example, suppose that $X \\subseteq \\mathbf{P}^2$ consists of the three coordinate points, so that $I_X = (xy, yz, zx) \\subseteq \\mathbf{C}[x,y,z]$. The Corollary guarantees that $I_X^a$ and $I_X^{(a)}$ agree in degrees $\\ge 2a + 2$, whereas in reality $\\textnormal{sat. \\!deg} (I_X^a) = 2a$. So here again the statement is asymptotically but not precisely sharp.\n\n\nIn the case of finite sets, results of Geramita-Gimigliano-Pitteloud \\cite{GGP}, Chandler \\cite{Chandler} and Sidman \\cite{Sidman} provide an alternative bound that is often best-possible. Recall that a scheme $X \\subseteq \\mathbf{P}^r$ is said to be $m$-regular in the sense of Castelnuovo--Mumford if its ideal sheaf $\\mathcal{I}_X \\subseteq \\mathcal{O}_{\\mathbf{P}^r}$ satisfies the vanishings:\n\\[\n\\HH{i}{\\mathbf{P}^r}{\\mathcal{I}_X(m-i)} \\ = \\ 0 \\ \\ \\text{ for \\ } i > 0. \n\\]\nThis is equivalent to asking that $I_X$ be generated in degrees $\\le m$, that the first syzygies among minimal generators of $I_X$ appear in degrees $\\le m+1$, the second syzygies in degrees $\\le m+2$, and so on.\\footnote{For saturated ideals, Castelnuovo--Mumford regularity of $I_X$ agrees with an algebraic notion of regularity introduced by Eisenbud and Goto \\cite{Eisenbud.Goto} that we propose to call \\textit{arithmetic regularity}. An arbitrary ideal $J \\subseteq S$ is arithmetically $m$-regular if and only if $\\satt{J}$ is $m$-regular and $\\textnormal{sat. \\!deg}(J) \\le m$. Given that we are interested in establishing bounds on saturation degree, unless otherwise stated we always refer to regularity in the geometric sense.}\nThe authors just cited show that if $X \\subseteq \\mathbf{P}^r$ is an $m$-regular finite set, then\n\\[\n\\textnormal{sat. \\!deg} (I_X^a) \\ \\le \\ am. \n\\]\n This is optimal for the example of the three coordinate points in $\\mathbf{P}^2$. \n \n Our second main result asserts that the same statement holds when $\\dim X = 1$. \n\\begin{theoremalpha} \\label{Regularity.Saturation.Bound.Curves}\nLet $X \\subseteq \\mathbf{P}^r$ be a smooth $m$-regular curve. Then\n\\[\n\\big( I_X^a\\big)_t \\ = \\ \\big( I_X^{(a)}\\big)_t \\ \\ \\text{for } \\ t \\, \\ge \\, a m. \n\\]\n\\end{theoremalpha}\n\\noindent In fact, for the saturation bound it suffices that the curve $X$ be reduced. The statement is optimal (for all $a$) for instance when $X \\subseteq \\mathbf{P}^4$ is a rational normal curve. \nWe also show that if $X \\subseteq \\mathbf{P}^r$ is a reduced surface, then $\\textnormal{reg}(\\mathcal{I}_X^a) \\le a \\cdot \\textnormal{reg}(\\mathcal{I}_X)$. \nWe do not know any examples where the analogous statements fail for smooth varieties of higher dimension. \n\nReturning to the setting of Theorem \\ref{Intro.Sat.Deg.Thm}, the first and third authors showed with Bertram some years ago \\cite{BEL} that if $X \\subseteq \\mathbf{P}^r$ is a smooth complex projective variety of codimension $e$ cut out as a scheme by homogeneous polynomials of degrees $d_0 \\ge \\ldots \\ge d_p$, then $\\mathcal{I}_X^a$ is $(ad_0 + d_1 + \\ldots + d_{e-1} -e)$-regular in the sense of Castelnuovo-Mumford. Note however that this does not address the questions of saturation required to control the arithmetic (Eisenbud--Goto) regularity of $I_X^a$.\\footnote{In particular, the proof of Proposition 2.2 in \\cite{AV} seems to be erroneous.} In fact, one can view Theorem \\ref{Intro.Sat.Deg.Thm} as promoting the results of \\cite{BEL} to statements about arithmetic regularity:\n \\begin{corollaryalpha} Assume that $J \\subseteq S$ satisfies the hypotheses of Theorem \\ref{Intro.Sat.Deg.Thm}. Then\n \\[ \\textnormal{arith. \\!reg}(J^a) \\ \\le \\ ad_0 + (d_1 + \\ldots + d_r -r).\\]\n \\end{corollaryalpha}\n\\noindent \nIt is known (\\cite{Kod}, \\cite{CHK}) that if $J \\subseteq S$ is an arbitrary homogeneous ideal then \\[ \\textnormal{arith. \\!reg}(J^a) \\ = \\ ad + b \\ \\text{ \\ when } \\ a \\gg 0, \\] where $d$ is the maximal degree needed to generate a reduction of $J$ -- which coincides with the generating degree of $J$ when it is equigenerated -- and $b$ is some constant. However computing the constant term $b$ has proven elusive, and the Corollary gives a bound in the case at hand. \n\nThe proofs of these results revolve around using complexes of sheaves to study the image in $\\HHHH{*}{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a} = \\satt{(I_X^{a})}$ of the powers of the ideal spanned by generators of $I_X$ or $J$: this approach was inspired in part by geometrizing the arguments of Cooper and coauthors for codimenson two subvarieties in \\cite{Cooper+}. Specifically, suppose that\n\\[ \\varepsilon : U_0 \\, =_{\\text{def}} \\, \\oplus \\, \\mathcal{O}_{\\mathbf{P}^r}(-d_i) \\longrightarrow \\mathcal{I}_X \\]\nis the surjective map of sheaves determined by generators of $I_X$ or $J$. If $X$ is $m$-regular, then this sits in an exact complex $U_\\bullet$ of bundles:\n\\[\n0 \\longrightarrow U_{r-1} \\longrightarrow U_{r-2} \\longrightarrow \\ldots \\longrightarrow U_1 \\longrightarrow U_0 \\overset{\\varepsilon} \\longrightarrow \\mathcal{I}_X \\longrightarrow 0\n\\] where $\\textnormal{reg}(U_i) \\le m + i$. Weyman \\cite{Weyman} (see also \\cite{Tchernev}) constructs a new complex $L_\\bullet = \\textnormal{Sym}^a(U_\\bullet)$ that takes the form\n\\[\n\\ldots \\longrightarrow L_2 \\longrightarrow L_1 \\longrightarrow S^a(U_0) \\longrightarrow \\mathcal{I}_X^a \\longrightarrow 0\n\\]\nwhere $\\textnormal{reg}(L_i) \\le am + i$. This complex is exact only off $X$, but as in \\cite{GLP} when $\\dim X = 1$ one can still read off the surjectivity of \n\\[\n\\HH{0}{\\mathbf{P}^r}{S^a (U_0)(t)} \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \n\\]\nfor $t \\ge am$. This gives Theorem \\ref{Regularity.Saturation.Bound.Curves}. \n\nTurning to Theorem \\ref{Intro.Sat.Deg.Thm}, a natural idea is to start with the Koszul complex\n\\[ \\ldots \\longrightarrow \\Lambda^3 U_0 \\longrightarrow \\Lambda^2 U_0 \\longrightarrow U_0 \\longrightarrow \\mathcal{I}_X \\longrightarrow 0.\\]\nAs established by Buchsbaum--Eisenbud \\cite{Buchsbaum.Eisenbud}, this determines a new complex\n\\[\n\\ldots \\longrightarrow S^{a, 1^2}(U_0) \\longrightarrow S^{a,1}(U_0) \\longrightarrow S^a(U_0) \\longrightarrow \\mathcal{I}_X^a \\longrightarrow 0, \\tag{*}\n\\]\nwhere $S^{a, 1^k}(U_0)$ denotes the Schur power of $U_0$ corresponding to the Young diagram $(a, 1^{k})$. We observe that \n\\[ \\textnormal{reg} \\big(S^{a, 1^{i}}(U_0) \\big) \\ \\le \\ ad_0 + d_1 + \\ldots + d_i, \\]\nso if (*) were exact then the statement of the Theorem would follow immediately. Unfortunately (*) is exact only if $X$ is a complete intersection, but by blowing up $X$ this construction yields an exact complex whose cohomology groups one can control with some effort. At the end of the day, the computation boils down to using Kodaira--Nakano vanishing on $X$ to prove \na vanishing statement for symmetric powers of the normal bundle to $X$ in $\\mathbf{P}^r$:\n\\begin{propositionalpha} \\label{NB.Vanishing.Prop}\nLet $X \\subseteq \\mathbf{P}^r$ be a smooth complex projective variety, and denote by $N = N_{X\/\\mathbf{P}^r}$ the normal bundle to $X$ in $\\mathbf{P}^r$. Then\n\\[ \\HHH{i}{X}{S^k N \\otimes \\det N \\otimes \\mathcal{O}_X(\\ell) } \\ = \\ 0 \\ \\ \\text{ for } i > 0\n\\]\nand every $k \\ge 0$, $\\ell \\ge -r$. \n\\end{propositionalpha}\n\\noindent (Similar but slightly different vanishings were established by Schneider and Zintl in \\cite{Schneider.Zintl}.)\n We hope that some of these ideas may find other applications in the future.\\footnote{We remark that some of the auxiliary results appearing here -- for example the Proposition just stated -- were known to the first and third authors some years ago in connection with their work on \\cite{BEL}. However they were put aside in favor of the simpler arguments with vanishing theorems that eventually appeared in that paper. }\n \n The paper is organized as follows. The first section is devoted to Theorem \\ref{Regularity.Saturation.Bound.Curves}. We collect in \\S 2 some preliminary results towards the Macaulay-type bounds. Specifically, we discuss the Buchsbaum--Eisenbud powers of Koszul complexes, the computation of some push-forwards from a blowing-up, and Proposition \\ref{NB.Vanishing.Prop}. The proof of Theorem \\ref{Intro.Sat.Deg.Thm} occupies \\S 3. We work throughout over the complex numbers.\n \n We are grateful to Sankhaneel Bisui, David Eisenbud, Elo\\'isa Grifo and Claudia Miller for valuable remarks and correspondence.\n \n \n\n\n\n\\numberwithin{equation}{section}\n\\section{Saturation and regularity}\n\nThe present section is devoted to the proof of Theorem \\ref{Regularity.Saturation.Bound.Curves} from the Introduction.\n\nWe start with some general remarks. Let $X \\subseteq \\mathbf{P}^r$ be a complex projective variety or scheme, with ideal sheaf $\\mathcal{I}_X \\subseteq \\mathcal{O}_{\\mathbf{P}^r}$ and homogeneous ideal $I_X \\subseteq S$. \nDenote by $U_\\bullet$ the locally free resolution of $\\mathcal{I}_X$ obtained by sheafifying a minimal graded free resolution of $I_X$:\n\\begin{equation} \\label{m-reg.Resoln.I}\n0 \\longrightarrow U_{r} \\longrightarrow U_{r-1} \\longrightarrow \\ldots \\longrightarrow U_1 \\longrightarrow U_0 \\overset{\\varepsilon} \\longrightarrow \\mathcal{I}_X \\longrightarrow 0. \\end{equation}\n Thus \neach $U_i$ is a direct sum of line bundles, and we recover the original resolution as the the complex $\\HHHH{*}{0}{\\mathbf{P}^r}{U_\\bullet}$ obtained from $U_\\bullet$ by taking global sections of all twists. \n\nConsider now the surjective homomorphism of sheaves\n\\[ S^a(\\varepsilon) \\, : \\, S^a U_0 \\longrightarrow \\mathcal{I}_X^a . \\]\nFor any $t \\ge 0$ one has\n\\[ \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \\ = \\ \\left( \\satt{\\left( I_X^a \\right)} \\right)_t. \\]\nOn the other hand, the fact that $U_0$ is constructed from minimal generators of $I_X$ implies that \n\\[\n\\textnormal{Im} \\Big( \\HH{0}{\\mathbf{P}^r}{S^a (U_0) (t) } \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \\Big) \\ = \\ \\big( I_X^a)_t .\n\\]\nTherefore\n\\begin{lemma} \\label{Surjectivity.Suffices.Lemma} The degree $t$ pieces of $I_X^a$ and $\\satt{(I_X^a)}$ coincide if and only if the homomorphism\n\\[\n\\HH{0}{\\mathbf{P}^r}{S^a (U_0) (t) } \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \n\\]\ndetermined by $S^a(\\varepsilon)$ is surjective. \\qed\n\\end{lemma}\n\\noindent The plan is to study $S^a(\\varepsilon)$ by realizing it as the last map of a complex $S^a( U_\\bullet)$.\n\nSpecifically, consider a smooth variety $M$, a subvariety $X \\subseteq M$, and a locally free resolution $U_\\bullet$ of $\\mathcal{I}_X \\subseteq \\mathcal{O}_M$ as above:\n\\begin{equation} \\label{m-reg.Resoln.II}\n0 \\longrightarrow U_{r} \\longrightarrow U_{r-1} \\longrightarrow \\ldots \\longrightarrow U_1 \\longrightarrow U_0 \\overset{\\varepsilon} \\longrightarrow \\mathcal{I}_X \\longrightarrow 0. \\end{equation}\nAs explained by Weyman \\cite{Weyman} and Tchernev \\cite{Tchernev}, $U_\\bullet$ determines for fixed $a \\ge 1$ a new complex $L_\\bullet = S^a(U_\\bullet)$ having the shape\n\\begin{equation} \\label{Weyman.Complex.1}\n\\xymatrix @C=18pt{\n\\ldots \\ar[r] & L_4 \\ar[r] &L_3 \\ar[r] & { \\begin{matrix} S^{a-2}U_0 \\otimes \\Lambda^2 U_1 \\\\ \\oplus \\\\ S^{a-1}U_0 \\otimes U_2 \\end{matrix}} \\ar[r] &S^{a-1}U_0 \\otimes U_1 \\ar[r] & S^aU_0\\ar[r]& \\mathcal{I}_X^a \\ar[r] & 0.\n}\n\\end{equation}\nThe last map on the right is $S^a(\\varepsilon)$, and the homomorphism $S^{a-1}U_0 \\otimes U_1 \\longrightarrow S^a U_0$ is the natural one arising as the composition\n \\[\n S^{a-1}U_0 \\otimes U_1 \\longrightarrow S^{a-1}U_0 \\otimes U_0 \\longrightarrow S^a U_0.\n \\] The $L_i$ are determined by setting\n \\begin{equation} \\label{Weyman.Complex.2}\n C^k (U_j) \\ = \\ \\begin{cases} \\ S^k U_j \\ & \\text{if $j$ is even} \\\\ \\ \\Lambda^k U_j \\ & \\text{if $j$ is odd} \\end{cases},\n \\end{equation}\nand then taking\n\\begin{equation} \\label{Weyman.Complex.3}\nL_i\\ = \\ \\bigoplus_{\n\\substack{k_0 + \\ldots + k_r = a \\\\ k_1 + 2k_2 + \\ldots + rk_r = i}} C^{k_0}(U_0) \\otimes C^{k_1}(U_1) \\otimes \\ldots \\otimes C^{k_r}(U_r).\n\\end{equation}\n\nIt follows from \\cite[Theorem 1]{Weyman} or \\cite[Theorem 2.1]{Tchernev} that:\n\\begin{equation} \\label{Exact.Away.From.X.Equation}\n\\text{The complex } \\eqref{Weyman.Complex.1}\n \\text{ is exact away from $X$}. \n\\end{equation}\nIn general one does not expect exactness at points of $X$, but when $X$ is smooth the right-most terms at least are well-behaved:\n\\begin{lemma} \\label{Right.Hand.Exactness.Weyman.Complex}\nAssume that $X$ is non-singular. Then the sequence\n\\[ S^{a-1}U_0 \\otimes U_1 \\longrightarrow S^a U_0 \\longrightarrow \\mathcal{I}_X^a \\longrightarrow 0\\]\nis exact. \n\\end{lemma}\n\\begin{proof} The question being local, we can work over the local ring $\\mathcal{O} = \\mathcal{O}_{M,x}$ of $M$ at a point $x \\in X$. Since $X$ is smooth, $\\mathcal{I} = \\mathcal{I}_{X, x} \\subseteq \\mathcal{O}$ is generated by a regular sequence of length $e = \\textnormal{codim} \\, X$. Thus $\\mathcal{I}$ has a minimal presentation \n\\[ \\Lambda^2 \\mathcal{U} \\longrightarrow \\mathcal{U} \\longrightarrow \\mathcal{I} \\longrightarrow 0 \n\\]\ngiven by the beginning of a Koszul complex, where $\\mathcal U = \\mathcal{O}^e$ is a free module of rank $e$. Here one checks by hand the exactness of \n\\[\nS^{a-1}\\mathcal{U} \\otimes \\Lambda^2 \\mathcal{U} \\longrightarrow S^a \\mathcal{U} \\longrightarrow \\mathcal{I}^a \\longrightarrow 0.\n\\] \n(Compare Proposition \\ref{Koszul.Complex.Power}\n below.) An arbitrary free presentation of $\\mathcal{I}$ then has the form\n\\[ \\Lambda^2 \\mathcal{U} \\oplus \\mathcal{A} \\oplus \\mathcal{B} \\longrightarrow \\mathcal{U} \\oplus \\mathcal{A} \\longrightarrow \\mathcal{I} \\longrightarrow 0,\\]\nwhere $\\mathcal{A}$ is a free module mapping to zero in $\\mathcal{I}$, $\\mathcal{B}$ is a free module mapping to zero in $\\mathcal{U} \\oplus \\mathcal{A}$, and the left-hand map is the identity on $\\mathcal{A}$. It suffices to verify the exactness of\n\\[\nS^{a-1}\\big( \\mathcal{U} \\oplus \\mathcal{A} \\big) \\otimes \\big( \\Lambda^2 \\mathcal{U} \\oplus \\mathcal{A} \\big) \\longrightarrow S^a \\big( \\mathcal{U} \\oplus \\mathcal{A}\\big) \\longrightarrow \\mathcal{I}^a \\longrightarrow 0, \n\\]\nand this is clear upon writing $S^a \\big( \\mathcal{U} \\oplus \\mathcal{A}\\big)=S^a \\mathcal{U} \\, \\oplus \\, \\mathcal{A} \\otimes \nS^{a-1}\\big( \\mathcal{U} \\oplus \\mathcal{A} \\big)$. \\end{proof}\n\n\nWith these preliminaries out of the way, we now prove (a slight strengthening of) Theorem \\ref{Regularity.Saturation.Bound.Curves} from the Introduction.\n\\begin{theorem} \\label{Reduced.Curve.Theorem} Let $X \\subseteq \\mathbf{P}^r$ be a reduced $($but possibly singular$)$ curve, and assume that $X$ is $m$-regular in the sense of Castelnuovo--Mumford. Denote by $I_X \\subseteq S$ the homogeneous ideal of $X$. Then\n\\[\n\\textnormal{sat. \\!deg}( I_X^a) \\ \\le \\ a m.\n\\]\n\\end{theorem}\n\\begin{proof}\nThe $m$-regularity of $X$ means that we can take a resolution $U_\\bullet$ of $\\mathcal{I}_X$ as in \\eqref{m-reg.Resoln.I} where $U_i$ is a direct sum of line bundles of degrees $\\ge -m -i$, ie $\\textnormal{reg}(U_i) \\le m + i$. Consider the resulting Weyman complex $L_\\bullet = S^a(U_\\bullet)$:\n\\[\n\\longrightarrow L_3 \\longrightarrow L_2 \\longrightarrow L_1 \\longrightarrow L_0 \\longrightarrow \\mathcal{I}_X^a \\longrightarrow 0, \\tag{*} \\]\nthe last map being the surjection $S^a(\\varepsilon) : L_0 = S^a U_0 \\longrightarrow \\mathcal{I}_X $. In view of Lemma \\ref{Surjectivity.Suffices.Lemma}, the issue is to establish the surjectivity of the homomorphism\n\\[ \\HH{0}{\\mathbf{P}^r}{L_0(t)} \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a(t)} \\tag{**} \\]\nfor $t \\ge am$. To this end, observe first from \\eqref{Weyman.Complex.2} and \n\\eqref{Weyman.Complex.3} that\n\\[ \\textnormal{reg}(L_i) \\ \\le \\ am + i. \\]\nConsider next the homology sheaves $\\mathcal{H}_i = \\mathcal{H}_i(L_\\bullet \\longrightarrow \\mathcal{I}_X^a)$ of the augmented complex (*). (So for $i = 0$ we understand $\\mathcal{H}_0 = \\ker ( L_0 \\longrightarrow \\mathcal{I}_X^a) \/ \\textnormal{Im}(L_1 \\longrightarrow L_0).)$ Thanks to \\eqref{Exact.Away.From.X.Equation}, these are all supported on the one-dimensional set $X$. Moreover it follows from Lemma \\ref{Right.Hand.Exactness.Weyman.Complex} that $\\mathcal{H}_0$ is supported on the finitely many singular points of $X$. Therefore the required surjectivity (**) is a consequence of the first statement of the following Lemma.\n \\end{proof}\n \n \\begin{lemma} Consider a complex $L_\\bullet$ of coherent sheaves on $\\mathbf{P}^r$ sitting in a diagram\n \\begin{equation}\\label{Chopping.Lemma.Eqn} \\ldots \\longrightarrow L_3 \\longrightarrow L_2 \\longrightarrow L_1 \\longrightarrow L_0\\overset{\\varepsilon} \\longrightarrow \\mathcal{F} \\longrightarrow 0, \\end{equation}\n and denote by $\\mathcal{H}_i = \\mathcal{H}_i(L_\\bullet \\longrightarrow \\mathcal{F})$ the $i^{\\text{th}}$ homology sheaf of the augmented complex \\eqref{Chopping.Lemma.Eqn}.\\footnote{So as above, the group of zero-cycles used to compute $\\mathcal{H}_0$ is $\\ker (\\varepsilon)$.}\nAssume that $\\varepsilon$ is surjective, and let $p$ be an integer with the property that $L_i$ is $(p + i)$-regular for every $i$. \n\\begin{enumerate}\n\\item [$(i)$] If each $\\mathcal{H}_i$ is supported on a set of dimension $\\le i$, then the homomorphism\n\\[ \\HH{0}{\\mathbf{P}^r}{L_0(t) } \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{F}(t)} \\]\nis surjective for $t \\ge p$. \n\\vskip 5pt\n\\item[$(ii)$] If each $\\mathcal{H}_i$ is supported on a set of dimension $\\le i + 1$, then $\\mathcal{F}$ is $p$-regular. \n\\end{enumerate}\n \\end{lemma}\n\\begin{proof}\n This is established by chopping $L_\\bullet$ into short exact sequences in the usual way and chasing through the resulting diagram. (Compare \\cite[B.1.2, B.1.3]{PAG}, but note that the sheaf $\\mathcal{H}_0$ there should refer to the augmented complex, as above.) \\end{proof}\n\nWe conclude this section by observing that the same argument proves that Castelnuovo--Mumford regularity of surfaces behaves submultiplicatively in powers. For curves, this has been known for some time \\cite{Chandler}, \\cite{Sidman}.\n\\begin{proposition}\nLet $X \\subseteq \\mathbf{P}^r$ be a reduced $($but possibly singular$)$ surface, and denote by $\\mathcal{I}_X \\subseteq \\mathcal{O}_{\\mathbf{P}^r}$ the ideal sheaf of $X$. If $\\mathcal{I}_X$ is $m$-regular, then $\\mathcal{I}_X^a$ is $am$-regular.\n\\end{proposition}\n\\begin{proof}[Sketch of Proof.] One argues just as in the proof of Theorem \\ref{Reduced.Curve.Theorem}, reducing to statement (ii) of the previous Lemma. \n\\end{proof}\n\n\n\n\n\n\n\\newcommand{\\Schur}[2]{S^{{#1},1^{#2}}}\n\n\\section{Macaulay-type bounds: preliminaries}\n\nThis section is devoted to some preliminary results that will be used in the proof of Theorem \\ref{Intro.Sat.Deg.Thm} from the Introduction. In the first subsection, we discuss symmetric powers of a Koszul complex. The second is devoted to the computation of some direct images from a blow-up. Finally \\S \\ref{Vanishing.Theorem.Normal.Bundles.Subsection} gives the proof of Proposition \\ref{NB.Vanishing.Prop} form the Introduction. \n\n\n\\subsection{Powers of Koszul complexes} \\label{Powers.of.Koszul.Subsection}\n\n\n\nIn this subsection we review the construction of symmetric powers of a Koszul complex. In the local setting this (and much more) appears in the paper \\cite{Buchsbaum.Eisenbud} of Buchsbaum and Eisenbud, and it was revisited by Srinivasan in \\cite{Srinivasan}. However for the convenience of the reader we give here a quick sketch of the particular facts we require. We continue to work over the complex numbers.\n\nLet $M$ be a smooth algebraic variety, and let $V$ be a vector bundle of rank $e$ on $M$. Fix integers $a, k \\ge 1$. We denote by $\\Schur{a}{k-1}(V)$ the Schur power of $V$ corresponding to the partition $(a, 1, \\ldots, 1)$ ($k-1$ repetitions of $1$). It follows from Pieri's rule that\n\\begin{equation} \\label{Schur.Equation}\n\\begin{aligned}\\Schur{a}{k-1}(V) \\ &= \\ \\ker \\Big( \\Lambda^{k-1}V \\otimes S^a V \\longrightarrow \\Lambda^{k-2}V \\otimes S^{a+1}V \\Big) \\\\ &= \\ \\textnormal{im} \\Big( \\Lambda^k V \\otimes S^{a-1}V \\longrightarrow \\Lambda^{k-1} V \\otimes S^a V \\Big). \n\\end{aligned}\n\\end{equation}\n\n\\begin{remark} [Properties of $\\Schur{a}{k-1}(V)$] \\label{Properties.of.Schur.Power} We collect some useful observations concerning this Schur power.\n\\begin{enumerate}\n\\item [(i).] If $ k = 1$ then $\\Schur{a}{k-1}(V) = S^aV$, while if $a = 1$ then $\\Schur{a}{k-1}(V) = \\Lambda^k V$. Moreover\n\\[ \\Schur{a}{k-1}(V) \\, = \\, 0 \\ \\ \\text{ when } k > \\rk V. \\]\n\\vskip 5pt\n\\item[(ii).] The bundle $\\Schur{a}{k-1}(V)$ is actually a summand of $S^{a-1}V \\otimes \\Lambda^k V$. In fact, Pieri shows that\n\\[\nS^{a-1}V \\otimes \\Lambda^k V \\ = \\ \\Schur{a}{k-1}(V)\\, \\oplus \\, \\Schur{a-1}{k}(V).\n\\]\n\\vskip 5pt\n\\item[(iii).] If $L$ is a line bundle on $M$, then it follows from \\eqref{Schur.Equation} or (ii) that \n\\[ \\Schur{a}{k-1} ( V \\otimes L ) \\ = \\ \\Schur{a}{k-1} ( V) \\, \\otimes \\, L^{\\otimes a + k -1}.\\]\n\\item[(iv).] Suppose that $M = \\mathbf{P}^r$ and \n\\[ V \\ = \\ \\mathcal{O}_{\\mathbf{P}^r}(-d_0 ) \\oplus \\, \\ldots \\, \\oplus \\, \\mathcal{O}_{\\mathbf{P}^r}(-d_p) \\]\nwith $\\ d_0 \\ge \\ldots \\ge d_p.$ Then it follows from (ii) that $\\Schur{a}{k-1}(V)$ is a direct sum of line bundles of degrees $\\ge \\, -(ad_0 + d_1 + \\ldots + d_{k-1})$, and moreover a summand of this degree appears. In other words,\n\\[ \\textnormal{reg} \\big( \\, \\Schur{a}{k-1}(V) \\, ) \\ = \\ ad_0 + d_1 + \\ldots + d_{k-1}.\\]\n\\end{enumerate}\n\\end{remark}\n\nOne can also realize $\\Schur{a}{k-1}(V)$ geometrically, \\`a la Kempf \\cite{Kempf}.\n\\begin{lemma} \\label{Kempf.Type.Lemma} \nLet $ \\pi : \\mathbf{P}(V) \\longrightarrow M $ be the projective bundle of one-dimensional quotients of $V$, and denote by $F$ the kernel of the canonical quotient $\\pi^* V \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}(1)$, so that $F$ sits in the short exact sequence\n\\[\n0 \\longrightarrow F \\longrightarrow \\pi^* V \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}(1) \\longrightarrow 0 \\tag{*}\n\\]\nof bundles on $\\mathbf{P}(V)$. Then\n\\[ \\Schur{a}{k-1}(V) \\ = \\ \\pi_* \\Big( \\, \\Lambda^{k-1}F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\, \\Big). \\]\n\\end{lemma} \n\\begin{proof} In fact, (*) gives rise to a long exact sequence\n\\small\n\\[\n0 \\longrightarrow \\Lambda^{k-1} F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\Lambda^{k-1} (\\pi^* V) \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\Lambda^{k-2} (\\pi^* V) \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a+1) \\longrightarrow \\ldots \\ . \n\\]\n\\normalsize\nThe assertion follows from \\eqref{Schur.Equation} upon taking direct images.\n\\end{proof}\n \n Now suppose given a map of bundles\n \\begin{equation} \\label{cosection}\n \\varepsilon : V \\longrightarrow \\mathcal{O}_M \\end{equation}\nwhose image is the ideal sheaf $\\mathcal{I} \\subseteq \\mathcal{O}_M$ of a subscheme $Z \\subseteq X$: equivalently, $\\varepsilon$ is dual to a section $\\mathcal{O}_M \\longrightarrow V^*$ whose zero-scheme is $Z$. We allow the possibility that $\\varepsilon$ is surjective, in which case $\\mathcal{I} = \\mathcal{O}_M$ and $Z = \\varnothing$. \n\n\nIf $Z$ has the expected codimension $e = \\rk(V)$, then $\\mathcal{I}$ is resolved by the Koszul complex associated to $\\varepsilon$. The following result of Buchsbaum and Eisenbud gives the resolution of powers of $\\mathcal{I}$.\n\\begin{proposition} [{\\cite[Theorem 3.1]{Buchsbaum.Eisenbud}, \\cite[Theorem 2.1]{Srinivasan}}] \\label{Power.Koszul.Complex.Proposition}\nFix $a \\ge 1$. Then $\\varepsilon$ determines a complex \n\\begin{equation} \\label{Koszul.Complex.Power}\n\\xymatrix@C=30pt{\n\\ldots \\ar[r] &\\Schur{a}{2}(V) \\ar[r] &S^{a,1}(V) \\ar[r] &S^a V \\ar[r]^{S^a(\\varepsilon)} & \\mathcal{I}^a \\ar[r] & 0 \n}\n\\end{equation}\nof vector bundles on $M$.\nThis complex is exact provided that either $\\varepsilon$ is surjective, or that $Z$ has codimension $= \\rk(V)$. \n\\end{proposition}\n\\noindent Observe from \\ref{Properties.of.Schur.Power} (i) that this complex has the same length as the Koszul complex of $\\varepsilon$. \n\n\\begin{proof} Returning to the setting of Lemma \\ref{Kempf.Type.Lemma}, denote by $\\tilde{\\varepsilon} : F \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}$ the composition of the inclusion $F \\hookrightarrow \\pi^*V$ with $\\pi^*\\varepsilon : \\pi^* V \\longrightarrow \\pi^* \\mathcal{O}_M$. The zero-locus of $\\tilde{\\varepsilon}$ defines the natural embedding of $\\mathbf{P}(\\mathcal{I})$ in $\\mathbf{P}(V)$. Now consider the Koszul complex of $\\tilde \\varepsilon$. After twisting by $\\mathcal{O}_{\\mathbf{P}(V)}(a)$ this has the form:\n\\[\n\\ldots \\longrightarrow \\Lambda^2 F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow F \\otimes \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\mathcal{O}_{\\mathbf{P}(V)}(a) \\longrightarrow \\mathcal{O}_{\\mathbf{P}(\\mathcal{I})}(a) \\longrightarrow 0. \\tag{*}\n\\]\nIn view of Lemma \\ref{Kempf.Type.Lemma}, \\eqref{Koszul.Complex.Power} arises by taking direct images. If $\\varepsilon$ is surjective, or defines a regular section of $V^*$, then the Koszul complex (*) is exact. Since the higher direct images of all the terms vanish, (*) pushes down to an exact complex. Furthermore, in this case $\\pi_* \\mathcal{O}_{\\mathbf{P}(\\mathcal{I})}(a) = \\mathcal{I}^a$ (cf \\cite[Theorem IV.2.2]{Fulton.Lang}), and the exactness of \\eqref{Koszul.Complex.Power} follows. \\end{proof}\n\n\n\\begin{example} [Macaulay's Theorem]\nSuppose as in the Introduction that $f_0, \\ldots, f_p \\in \\mathbf{C}[x_0, \\ldots, x_r]$ are homogeneous polynomials of degrees $d_0 \\ge \\ldots \\ge d_p$ that generate a finite colength ideal $J$. This gives rise to a surjective map\n\\[ V \\ = \\ \\oplus \\, \\mathcal{O}_{\\mathbf{P}^r} (-d_i) \\longrightarrow \\mathcal{O}_{\\mathbf{P}^r} \\longrightarrow 0\\]\nof bundles on projective space. Keeping in mind Remark \\ref{Properties.of.Schur.Power} (iv), Macaulay's statements \\eqref{Macaulay.Eqn.1}\n and \\eqref{Macaulay.Eqn.2}\nfollow by looking at the cohomology of the resulting complex \\eqref{Koszul.Complex.Power}. When $p = r$ this complex has length $r+1$, so one can also read off the non-surjectivity of \n\\[ \\HH{0}{\\mathbf{P}^r}{S^a V (t)} \\longrightarrow \\HH{0}{\\mathbf{P}^r}{\\mathcal{O}_{\\mathbf{P}^r}(t)} \\]\nwhen $t < ad_0+ d_1 + \\ldots + d_r -r$. \n\\end{example}\n\n\\begin{example} [Complete intersection ideals] Suppose that $Z\\subseteq \\mathbf{P}^r$ is a complete intersection of dimension $\\ge 0$. Applying Theorem \\ref{Koszul.Complex.Power}\nto the Koszul resolution of its homogeneous ideal $I_Z$, one sees that $I_Z^a$ is saturated for every $a\\ge 1$. This is a result of Zariski.\n\\end{example}\n\n\\subsection{Push-forwards from a blowing up}\n\\label{Pushfowards.from.Blowup.Subsection}\n\n \nWe compute here the direct images of multiples of the exceptional divisor under the blowing-up of a smooth subvariety.\n\nConsider then a smooth variety $M$ and a non--singular subvariety $X \\subseteq M$ having codimension $e \\ge 2$ and ideal sheaf $\\mathcal{I} = \\mathcal{I}_X \\subseteq \\mathcal{O}_M$. We consider the blowing-up \n\\[ \\mu : M^\\prime = \\text{Bl}_X(M) \\longrightarrow M \\]\nof $M$ along $X$. Write $\\mathbf{E} \\subseteq M^\\prime$ for the exceptional divisor of $M^\\prime$, so that $\\mathcal{I} \\cdot \\mathcal{O}_{M^\\prime} = \\mathcal{O}_{M^\\prime}(-\\mathbf{E})$. \nWe recall that if $a>0$ then\n\\begin{equation} \\label{BU.Eqn.1}\n\\mu_* \\mathcal{O}_{M^\\prime}(-a\\mathbf{E}) \\ = \\ \\mathcal{I}^a \\ \\ \\text{and } \\ \\ R^j \\mu_* \\mathcal{O}_{M^\\prime}(-a\\mathbf{E}) \\, = \\, 0 \\ \\text{for } j > 0.\n\\end{equation}\n\nThe following Proposition gives the analogous computation for positive multiples of $\\mathbf{E}$.\n\\begin{proposition} \\label{Blowup.Pushforward.Proposition}\nFix $a > 0$. Then\n\\begin{equation} \\label{Pushforward.Ext.Equation}\nR^j \\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \\ = \\ \\mathcal{E}\\mathit{xt}^j_{\\mathcal{O}_M}\\Big( \\mathcal{I}^{a-e+1} \\, , \\, \\mathcal{O}_M \\Big).\\footnote{When $0 < a < e-1$ we take $\\mathcal{I}^{a-e+1} = \\mathcal{O}_M$.}\n\\end{equation}\nIn particular, $\\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) = \\mathcal{O}_M$, $R^j \\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) = 0$ if $j \\ne 0, e-1$, and\n\\[\nR^{e-1}\\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \\ = \\ \\mathcal{E}\\mathit{xt}^{e-1}_{\\mathcal{O}_M}\\big( \\mathcal{I}^{a-e+1} \\, , \\, \\mathcal{O}_M \\big). \\]\n\\end{proposition} \n\n\\begin{proof}[Proof of Proposition \\ref{Blowup.Pushforward.Proposition}]\nThis is a consequence of duality for $\\mu$, which asserts that\n\\[ \nR\\mu_* \\, R\\,\\mathcal{H}\\mathit{om}_{\\mathcal{O}_{M^\\prime}} \\big( \\mathcal{F} \\, , \\, \\omega_\\mu \\, \\big) \\ = \\ R\\, \\mathcal{H}\\mathit{om}_{\\mathcal{O}_M} \\big ( \\, R\\mu_* \\mathcal{F} \\, , \\mathcal{O}_M \\, \\big) \\tag{*}\n\\]\nfor any sheaf $\\mathcal{F}$ on $M^\\prime$, where $\\omega_\\mu$ denotes the relative dualizing sheaf for $\\mu$ (\\cite[(3.19) on page 86]{Huybrechts}). We apply this with \n\\[ \\mathcal{F} \\ = \\ \\mathcal{O}_{M^\\prime}\\big( \\, (e-1-a)\\mathbf{E}\\, ). \\]\nThen \n$R \\mu_* \\mathcal{F} = \\mathcal{I}^{a-e+1}$\nthanks to \\eqref{BU.Eqn.1} (and a direct computation when $0 < a < e-1$), and $\\omega_\\mu = \\mathcal{O}_{M^\\prime}\\big( (e-1)\\mathbf{E} \\big)$. Therefore the first assertion of the Proposition follows from (*). The vanishing of $\\mathcal{E}\\mathit{xt}^j_{\\mathcal{O}_M}(\\mathcal{I}^{a-e+1}, \\mathcal{O}_M)$ for $j \\ne 0, e-1$ follows from the perfection of powers of the ideal of a smooth variety (which in turn is a consequence eg of Proposition \\ref{Power.Koszul.Complex.Proposition}). \n\\end{proof}\n\n\n\\begin{remark} [Generalization to multiplier ideal sheaves] Let $\\mathfrak{b} \\subseteq \\mathcal{O}_M$ be an arbitrary ideal sheaf, and let $\\mu : M^\\prime \\longrightarrow M$ be a log resolution of $\\mathfrak{b}$, with $\\mathfrak{b} \\cdot \\mathcal{O}_{M^\\prime} = \\mathcal{O}_{M^\\prime}(-\\mathbf{E})$. A completely parallel argument shows that for $a > 0$:\n\\[\nR^j \\mu_* \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \\ = \\ \\mathcal{E}\\mathit{xt}^j_{\\mathcal{O}_M}\\big( \\MI{\\mathfrak{b}^a} \\, , \\, \\mathcal{O}_M \\big),\\]\nwhere $\\MI{\\mathfrak{b}^a}$ is the multiplier ideal of $\\mathfrak{b}^a$. The formula \\eqref{Pushforward.Ext.Equation} is a special case of this. \n\\end{remark}\n\n\n\n\\begin{corollary} \\label{Filtration.of.Push.Forwards}\nContinuing to work in characteristic zero, fix $a \\ge 1$ and denote by $N = N_{X\/M}$ the normal bundle to $X$ in $M$. If $a \\le e-1$, then \n\\[ R^{e-1} \\mu_* \\, \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \\ = \\ 0. \\] If $a \\ge e$, then\n$\nR^{e-1} \\mu_* \\, \\mathcal{O}_{M^\\prime}(a\\mathbf{E}) \n$\nhas a filtration with successive quotients\n\\[\nS^k N \\otimes \\det N \\ \\ \\text{ for } \\ 0 \\, \\le \\, k \\, \\le \\, a-e.\n\\]\n\\end{corollary}\n\\begin{proof} The first statement follows directly from the previous Proposition. For the second, \nrecall first that if $E$ is any locally free $\\mathcal{O}_X$-module, then -- $X$ being non-singular of codimension $e$ in $M$ -- \n\\[ \\mathcal{E}\\mathit{xt}_{\\mathcal{O}_M}^{e} \\big ( \\, E \\, , \\, \\mathcal{O}_M \\, \\big) \\ = \\ E^* \\otimes \\det N, \\]\nwhile all the other $\\mathcal{E}\\mathit{xt}^j$ vanish. The claim then follows from Proposition \\ref{Pushforward.Ext.Equation}\n using the exact sequences\n\\[ 0 \\longrightarrow \\mathcal{I}^{k+1} \\longrightarrow \\mathcal{I}^k \\longrightarrow S^k N^* \\longrightarrow 0 \\]\ntogether with the isomorphism $\\big (S^k(N^*)\\big)^* = S^k N$ valid in characteristic zero.\n \\end{proof}\n\n\n\\begin{remark}\nRecalling that $\\mathbf{E} = \\mathbf{P}(N^*)$, one can inductively prove the Corollary directly, circumventing Proposition \\ref{Pushforward.Ext.Equation}, by pushing forward the exact sequences\n\\[ 0 \\longrightarrow \\mathcal{O}_{M^\\prime}\\big((k-1)\\mathbf{E}\\big) \\longrightarrow \\mathcal{O}_{M^\\prime}\\big(k \n\\mathbf{E}\\big) \\longrightarrow \\mathcal{O}_{\\mathbf{E}}(k\\mathbf{E})\\longrightarrow 0. \\]\nHowever it seemed to us that the Proposition may be of independent interest. \n\\end{remark}\n\n\n\\subsection{A vanishing theorem for normal bundles}\n\\label{Vanishing.Theorem.Normal.Bundles.Subsection}\n\nThis final subsection is devoted to the proof of \n\\begin{proposition} \\label{Van.Thm.NB.Subsection.Statement}\nLet $X \\subseteq \\mathbf{P}^r$ be a smooth complex projective variety of dimension $n$, and denote by $N = N_{X\/\\mathbf{P}^r}$ the normal bundle to $X$. Then\n\\[ \\HHH{i}{X}{S^kN \\otimes \\det N \\otimes \\mathcal{O}_X(\\ell)} \\ = \\ 0 \n\\]\nfor all $i > 0$, $k\\ge 0$ and $\\ell \\ge -r$. \n\\end{proposition}\n\\noindent Here $\\mathcal{O}_X(k)$ denotes $\\mathcal{O}_{\\mathbf{P}^r}(k)|X$. We remark that similar statements were established by Schneider and Zintl in \\cite{Schneider.Zintl}, but this particular vanishing does not seem to appear there. Other vanishings for normal bundles played a central role in \\cite{SAD}. \n\n\\begin{proof} [Proof of Proposition \\ref{Van.Thm.NB.Subsection.Statement}]\nWe use the abbreviation $\\mathbf{P} = \\mathbf{P}^r$. Starting from the exact sequence $ 0 \\longrightarrow TX \\longrightarrow T\\mathbf{P}|X \\longrightarrow N \\longrightarrow 0$, we get a long exact sequence\n\\[ \\ldots \\longrightarrow S^{k-2}T\\mathbf{P}|X \\otimes \\Lambda^2 TX \\longrightarrow S^{k-1}T\\mathbf{P}|X \\otimes TX \\longrightarrow S^k T\\mathbf{P}|X \\longrightarrow S^k N \\longrightarrow 0. \\tag{*} \\]\nBy adjunction, $\\det N \\otimes \\mathcal{O}_X(\\ell) = \\omega_X \\otimes \\mathcal{O}_X(\\ell+r+1)$. So after twisting through by $\\det N \\otimes \\mathcal{O}_X(\\ell)$ in (*), we see that the Proposition will follow if we prove:\n\\[\n\\HHH{i}{X}{S^{k-j}T\\mathbf{P}|X \\otimes \\Lambda^j TX \\otimes \\omega_X \\otimes \\mathcal{O}_X(\\ell+ r + 1))} \\ = \\ 0 \\ \\ \\ \\text{for } \\ i \\, \\ge \\, j + 1 \\tag{**}\n\\]\nwhen $0 \\le j \\le k$ and $\\ell \\ge -r$. It follows from the Euler sequence that $S^m T\\mathbf{P}|X$ has a presentation of the form\n\\[ 0 \\longrightarrow \\oplus \\, \\mathcal{O}_X(m-1) \\longrightarrow \\oplus \\, \\mathcal{O}_X(m) \\longrightarrow S^m T\\mathbf{P} |X \\longrightarrow 0, \\]\nso for (**) it suffices in turn to verify that\n\\[ \\HH{i}{X}{\\Lambda^j TX \\otimes \\omega_X \\otimes \\mathcal{O}_X(\\ell_1)} \\ = \\ 0 \\] for $i \\ge j + 1$ and $\\ell_1 > 0$. But \n$ \\Lambda^j TX \\otimes \\omega_X = \\Omega^{n-j}_X$,\nso finally we're asking that\n\\[ \\HH{i}{X}{\\Omega^{n-j}_X \\otimes \\mathcal{O}_X(\\ell_1)} \\ = \\ 0 \\ \\ \\text{for } \\ i \\ge j + 1 \\ \\text{and } \\ell _1 >0,\\]\nand this follows from Nakano vanishing. \\end{proof}\n\n\\section{Proof of Theorem \\ref{Intro.Sat.Deg.Thm} }\n\nWe now turn to the proof of Theorem \\ref{Intro.Sat.Deg.Thm}\n from the Introduction.\n \nConsider then a non-singular variety $X \\subseteq \\mathbf{P}^r$ that is cut out as a scheme by hypersurfaces of degrees $d_0 \\ge \\ldots \\ge d_p$. Equivalently, we are given a surjective homomorphism of sheaves:\n\\[ \\varepsilon : U \\longrightarrow \\mathcal{I}_X \\ \\ \\text{,} \\ \\ U \\ = \\ \\oplus \\, \\mathcal{O}_{\\mathbf{P}^r}(-d_i). \\]\nLet \n$\\mu : \\mathbf{P}^\\prime = \\text{Bl}_X(\\mathbf{P}^r) \\longrightarrow \\mathbf{P}^r $ be the blowing up of $X$, with exceptional divisor $\\mathbf{E} \\subseteq \\mathbf{P}^\\prime$, so that $\\mathcal{I}_X \\cdot \\mathcal{O}_{\\mathbf{P}^\\prime} = \\mathcal{O}_{\\mathbf{P}^\\prime} (- \\mathbf{E})$. Write $H$ for the pull-back to $\\mathbf{P}^\\prime$ of the hyperplane class on $\\mathbf{P}^r$, and set $U^\\prime = \\mu^* U$. Thus on $\\mathbf{P}^\\prime$ we have a surjective map of bundles:\n\\begin{equation} \\label{Map.of.Bundles.on.Blowup}\n\\varepsilon^\\prime : U^\\prime \\longrightarrow \\mathcal{O}_{\\mathbf{P}^\\prime}(-\\mathbf{E}).\n\\end{equation}\nNoting that\n\\[ \\HHH{0}{\\mathbf{P}^\\prime}{\\mathcal{O}_{\\mathbf{P}^\\prime}( tH - a \\mathbf{E})} \\ = \\ \\HHH{0}{\\mathbf{P}^r}{\\mathcal{I}_X^a \\otimes\\mathcal{O}_{\\mathbf{P}^r}(t)},\n\\]\none sees as in Lemma \\ref{Surjectivity.Suffices.Lemma} that the question is to prove the surjectivity of \n\\begin{equation} \\label{Surjectivity.Required.for.Thm.A}\n\\HHH{0}{\\mathbf{P}^\\prime}{S^aU^\\prime\\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(tH)} \\longrightarrow \\HHH{0}{\\mathbf{P}^\\prime}{\\mathcal{O}_{\\mathbf{P}^\\prime}(tH- a \\mathbf{E})} \n\\end{equation}\nfor $t \\ge ad_0 + d_1 + \\ldots + d_r - r$. \n\nTo this end, we pass to the Buchsbaum--Eisenbud complex \\eqref{Koszul.Complex.Power} constructed from \\[ U^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(\\mathbf{E}) \\overset{\\varepsilon^\\prime} \\longrightarrow \\mathcal{O}_{\\mathbf{P}^\\prime} \\longrightarrow 0. \\]\nTwisting through by $\\mathcal{O}_{\\mathbf{P}^\\prime}(t H - a\\mathbf{E})$, we arrive at a long exact sequence of vector bundles on $\\mathbf{P}^\\prime$ having the form:\n\n\\vskip -10pt\n\\small\n\\begin{equation} \\label{Big.Complex.on.Blowup}\n\\xymatrix@C=9.5pt@R=12pt{\n\\ldots \\ar[r] &\\Schur{a}{2} U^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(t H + 2\\mathbf{E}) \\ar[r] \\ar@{=}[d]&S^{a,1} U^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(t H+ \\mathbf{E}) \\ar[r]\\ar@{=}[d] & S^aU^\\prime \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(t H) \\ar[r] \\ar@{=}[d] &\\mathcal{O}_{\\mathbf{P}^\\prime}(t H - a \\mathbf{E}) \\ar[r] &0. \\\\ & C_2 & C_1 & C_0\n}\n\\end{equation}\n\\normalsize\nWith indexing as indicated, the $i^{\\text{th}}$ term of this sequence is given by\n\\[\nC_i \\ = \\ \\Schur{a}{i}(U^\\prime) \\otimes \\mathcal{O}_{\\mathbf{P}^\\prime}(tH + i \\mathbf{E}). \n\\]\n\n\nIn order to establish the surjectivity \\eqref{Surjectivity.Required.for.Thm.A} it suffices upon chasing through \\eqref{Big.Complex.on.Blowup} to prove that\n\\begin{equation} \\label{Vanishing.Required.for.Thm.A}\n\\HH{i}{\\mathbf{P}^\\prime}{C_i} \\ = \\ 0 \\ \\ \\text{ for } \\ 1 \\le i \\le r\n\\end{equation}\nprovided that $t \\ge ad_0 + d_1 + \\ldots + d_r -r$. But now recall (Remark \\ref{Properties.of.Schur.Power}) that if $i \\le r$ then $\\Schur{a}{i}(U^\\prime)$ is a sum of line bundles $\\mathcal{O}_{\\mathbf{P}^\\prime}(mH)$ with \n\\[\nm \\, \\ge \\, -ad_0 - d_i - \\ldots - d_i \\ge -ad_0 - d_1 - \\ldots - d_r.\n\\]\nHence when $t \\ge ad_0 + d_1 + \\ldots + d_r -r$, $C_i$ is a sum terms of the form\n\\[ \n\\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E}) \\ \\ \\text{with } \\ell \\ge -r.\n\\]\nTherefore \\eqref{Vanishing.Required.for.Thm.A}\n -- and with it Theorem \\ref{Intro.Sat.Deg.Thm} -- is a consequence of\n\\begin{proposition}\nIf $\\ell \\ge -r$, then\n\\[ \\HH{i}{\\mathbf{P}^\\prime}{\\mathcal{O}_{\\mathbf{P}^\\prime}{(\\ell H + i \\mathbf{E})} } \\ = \\ 0 \\ \\text{ for \\ } i > 0.\\]\n\\end{proposition}\n\\begin{proof}\nThanks to the Leray spectral sequence, it suffices to show:\n\\[\n\\HH{j}{\\mathbf{P}^r}{R^k \\mu_* \\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E})} \\ = \\ 0 \\ \\ \\text{when } j + k = i > 0. \\tag{*}\n\\]\nFor $ k = 0$, observe that $\\mu_* \\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E}) = \\mathcal{O}_{\\mathbf{P}^r}(\\ell)$, and these sheaves have no higher cohomology when $\\ell \\ge -r$. On the other hand, by Proposition \\ref{Pushforward.Ext.Equation} the only non-vanishing higher direct images are the $R^{e-1}\\mu_* \\mathcal{O}_{\\mathbf{P}^\\prime}(\\ell H + i \\mathbf{E})$, which do not appear when $i \\le e-1$. So (*) holds when $j = 0, k = e-1$. It remains to consider the case $k = e-1$ and $i \\ge e$, so $j = i - (e-1) > 0$. \nHere Corollary \\ref{Filtration.of.Push.Forwards}\n implies that the $R^{e-1}$ have a filtration with quotients\n \\[\n S^\\alpha N \\otimes \\det N \\otimes \\mathcal{O}_X(\\ell),\n \\]\n where as above $N = N_{X\/\\mathbf{P}^r}$ is the normal bundle to $X$ in $\\mathbf{P}^r$. But since we are assuming $\\ell \\ge -r$, Proposition \\ref{Van.Thm.NB.Subsection.Statement} guarantees that these sheaves have vanishing higher cohomology. This completes the proof. \\end{proof}\n\n\\begin{remark}\\label{Few.Equations}\nObserve that if $X$ is defined by $p < r$ equations, then the argument just completed goes through taking\n $ d_{p+1} = \\ldots = d_r = 0.$\n \\end{remark}\n\n\n\n\n %\n %\n\n %\n %\n\n \n \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}