On the notion of sequentially Cohen-Macaulay modules (Q2157924)

This nice paper surveys sequentially Cohen-Macaulay modules. Let \(k\) be a field. Let \(R\) be either a Noetherian local ring with maximal ideal \(\mathfrak{m}\) and residue field \(k\), or a standard graded \(k\)-algebra \(R=\underset{i\geq 0}\bigoplus R_i\), with \(R_0=k\) and \(\mathfrak{m}=\underset{i>0}\bigoplus R_i\). In the second case, every \(R\)-module is assumed to be \(\mathbb{Z}\)-graded. A finitely generated \(R\)-module \(M\) is called \textit{sequentially Cohen-Macaulay} if it has a filtration of submodules \[ 0=M_0\subsetneq M_1\subsetneq \dots \subsetneq M_r=M \] such that each quotient \(M_i/M_{i-1}\) is Cohen-Macaulay and \(\dim_R(M_i/M_{i-1})<\dim_R(M_{i+1}/M_i)\) for all \(i\). Such a filtration is called a \textit{sCM filtration} of \(M\). Let \(M\) be a \(d\)-dimensional finitely generated \(R\)-module. There is a unique filtration \[ 0=\delta_{-1}(M)\subseteq \delta_0(M)\subseteq \delta_1(M)\subseteq \dots \subseteq \delta_d(M)=M, \] where \(\delta_i(M)\) is the largest submodule of \(M\) of dimension less than or equal to \(i\) for all \(i\). The \(R\)-module \(M\) is called \textit{Cohen-Macaulay filtered} if for every \(0\leq i\leq d\), the module \(\delta_i(M)/\delta_{i-1}(M)\) is either zero or Cohen-Macaulay. Among other things, the authors provide new proofs for the following three essential results concerning sequentially Cohen-Macaulay \(R\)-modules. {Theorem (Schenzel):} A finitely generated \(R\)-module \(M\) is sequentially Cohen-Macaulay if and only if it is Cohen-Macaulay filtered. Moreover, if a finitely generated \(R\)-module \(M\) is sequentially Cohen-Macaulay, then its sCM filtration is unique. {Theorem (Peskine):} Let \(R\) be a Cohen-Macaulay ring of dimension \(n\) with canonical module \(\omega_R\) and \(M\) a finitely generated \(d\)-dimensional \(R\)-module. Then, the following are equivalent: \begin{itemize} \item[(i)] \(M\) is sequentially Cohen-Macaulay. \item[(ii)] \(\mathrm{Ext}_R^{n-i}(M,\omega_R)\) is either \(0\) or Cohen-Macaulay of dimension \(i\) for all \(0\leq i\leq d\). \item[(iii)] \(\mathrm{Ext}_R^{n-i}(M,\omega_R)\) is either \(0\) or Cohen-Macaulay of dimension \(i\) for all \(0\leq i\leq d-1\). \end{itemize} {Theorem:} Let \(M\) be a finitely generated graded \(R\)-module and \(M\cong F/U\) a free graded presentation of \(M\). Then \(M\) is sequentially Cohen-Macaulay if and only if \(\mathrm{Hilb}(\mathrm{H}^i_{\mathfrak{m}}(F/U))= \mathrm{Hilb}(\mathrm{H}^i_{\mathfrak{m}}(F/\mathrm{Gin}(U)))\) for all \(i\geq 0\).\\ Here, for an \(R\)-module \(N\), \(\mathrm{H}^i_{\mathfrak{m}}(N)\) denotes the \(i\)-th local cohomology module of \(N\) with respect to the ideal \(\mathfrak{m}\). The notation \(\mathrm{Hilb}\) stands for Hilbert function and \(\mathrm{Gin}(U)\) denotes the generic initial module of \(U\) with respect to the degree reverse lexicographic order.