On the cofinite modules with respect to an ideal of cohomological dimension not exceeding one (Q6634433)

All rings in this review are commutative with identity and all modules are unitary. Let \(R\) be a Noetherian ring with non-zero identity, \(I\) be an ideal of \(R\), and \(M\) be an \(R\)-module. Recall that for any non-negative integer \(i\), the \(i\)-th local cohomology module of \(M\) with respect to \(I\) is defined as \(\mathrm{H}_I^i(M)=\underrightarrow\lim_{n \geq 1}\mathrm{Ext}_R^i(R/I^n, M)\). The cohomological dimension of \(M\) with respect to \(I\), denoted by \(\mathrm{cd}(I, M)\) is defined as the supremum of all \(i\) such that \(\mathrm{H}_I^i(M) \neq 0\) provided that the supremum exists; otherwise \(- \infty\). Moreover, an \(R\)-module \(M\) is called \(I\)-cofinite if \(\mathrm{Supp}_R (M) \subseteqq\mathrm{V}(I)\), and \(\mathrm{Ext}_R^i(R/I, M)\) is a finite module for all \(i\). The authors introduce the notion \(\overline{\mathrm{Ass}}_{R}M\) as the set of all \(\mathfrak{p} \in \mathrm{Spec} R\) such that \(\mathrm{Ann}_R(0 :_M \mathfrak{p})=\mathfrak{p}\).\N\NIn the paper under review, the authors show that if \(R\) is a Noetherian complete local ring and \(I\) is an ideal of \(R\) satisfying \(\mathrm{cd}(I, R) \leq 1\), then for any \(I\)-cofinite \(R\)-module \(M\), \(\overline{\mathrm{Ass}}_{R}M\) is finite (See Theorem 2.14). Furthermore, for any ideal \(I\) of a Noetherian local ring \(R\) satisfying \(\mathrm{cd}(I, R) \leq 1\) and each \(I\)-cofinite \(R\)-module \(M\), the set of all maximal elements of \(\overline{\mathrm{Ass}}_{R}M \setminus\mathrm{V}(I)\) denoted by \(\Lambda_R(I, M)\) is a finite subset of \(\mathrm{V} (\mathrm{Ann}_RM)\) (See Theorem 3.3). Under the same hypotheses as Theorem 3.3, for each \(a \in I\), the \(R\)-module \((0 :_M a)\) is finitely generated exactly if \(a \notin \bigcup_{\mathfrak{p} \in \Lambda_R(I, M)} \mathfrak{p}\) (see Theorem 3.4). Finally, some corollaries are obtained about local cohomology modules. Several technical results have been proved to reach the main results.