Cofinite modules and cofiniteness of local cohomology modules (Q6589676)

All rings in this review are commutative with identity and all modules are unitary. Let \(R\) be a Noetherian ring with non-zero identity, \(\mathfrak{a}\) an ideal of \(R,\) \(M\) a finitely generated (or finite for short) \(R\)-module, \(X\) an arbitrary \(R\)-module, and \(n\) a non-negative integer. Recall that an \(\mathfrak{a}\)-torsion \(R\)-module \(X\) is said to be \(\mathfrak{a}\)-cofinite if \(\mathrm{Ext}^i_R(R/\mathfrak{a}, X)\) is a finite \(R\)-module for all \(i.\) Moreover, the module \(X\) is said to be an \(\mathrm{FD}_{<n}\) \(R\)-module if there exists a finite \(R\)-submodule \(X'\) of \(X\) such that \(\dim_R(X/X') < n.\) Finally, the authors call \(X\) to be an (\(\mathrm{FD}_{<n}\), \(\mathfrak{a}\))-cofinite \(R\)-module if \(X\) is an \(\mathfrak{a}\)-torsion \(R\)-module and \(\mathrm{Ext}^i_R(R/\mathfrak{a}, X)\) is an \(\mathrm{FD}_{<n}\) \(R\)-module for all \(i.\) For a subset \(A\) of \(\mathrm{Spec}(R),\) the set \(\{\mathfrak{p} \in A:\dim(R/\mathfrak{p}) \geq n\}\) is denoted by \(A_{\geq n}.\) On the basis of the importance of the following questions in local cohomology, the first author and his colleagues have provided affirmative answers to them under some conditions in the past years:\N\begin{itemize}\N\item[] Is \(\mathrm{H}^i_{\mathfrak{a}}(M)\) an \(\mathfrak{a}\)-cofinite \(R\)-module for all \(i\)?\N\item[] Is \(\mathrm{Ass}_R(\mathrm{H}^i_{\mathfrak{a}}(M))\) a finite set for all \(i\)?\N\end{itemize}\NIn the paper under review, the authors investigate about the following questions which are natural generalizations of the above-mentioned ones:\N\begin{itemize}\N\item[] Is \(\mathrm{H}^i_{\mathfrak{a}}(M)\) an (\(\mathrm{FD}_{<n}\), \(\mathfrak{a}\))-cofinite \(R\)-module for all \(i\)?\N\item[] Is \(\mathrm{Ass}_R(\mathrm{H}^i_{\mathfrak{a}}(M))_{\geq n}\) a finite set for all \(i\)?\N\end{itemize}\NThey give affirmative answers to these questions under the condition that \(\dim_R(M)\leq n+ 2\) (See Theorem 2.1 and Corollary 2.2). Moreover, it is shown that if \(\operatorname{H}^{i}_{\mathfrak{a}}(X)\) is an \(\operatorname{FD}_{< n+ 2}\) \(R\)-module for all \(i< \dim_R(X)- n\) and \(\operatorname{Ext}^{i}_{R}(R/\mathfrak{a}, X)\) is an \(\operatorname{FD}_{< n}\) \(R\)-module for all \(i\leq \dim_R(X)- n\), then \(\operatorname{Ext}^{i}_{R}(R/\mathfrak{a}, X)\) is an \(\operatorname{FD}_{< n}\) \(R\)-module for all \(i\) (See Theorem 2.5). As a consequence, it follows that if \(\operatorname{Ext}^{i}_{R}(R/\mathfrak{a}, X)\) is an \(\operatorname{FD}_{< n}\) \(R\)-module for all \(i\leq \dim_R(X)- n\), then the \(i\)th Bass number and the \(i\)th Betti number of \(X\) with respect to \(\mathfrak{p}\) are finite for every integer \(i\) and every prime ideal \(\mathfrak{p}\) of \(\operatorname{Var}(\mathfrak{a})_{\geq n}\). Finally, the authors prove that if \(\dim(R/\mathfrak{a})\leq 2\) and \(X\) is an \(\mathfrak{a}\)-torsion \(R\)-module such that \(\operatorname{Supp}_R(X)\cap \operatorname{Var}(\mathfrak{a})\cap \operatorname{Max}(R)\) is a finite set (e.g., \(R\) is semi-local) and \(\operatorname{Ext}^{i}_{R}(R/\mathfrak{a}, X)\) is a finite \(R\)-module for all \(i\leq 2\), then \(X\) is an \(\mathfrak{a}\)-cofinite \(R\)-module (See Theorem 2.11(a)). Some ordinary \(\mathfrak{a}\)-cofiniteness results for local cohomology modules \(\mathrm{H}^i_{\mathfrak{a}}(X)\) are concluded as well.