Some results on cofinite modules (Q2920233)

Let \(R\) be a commutative Noetherian ring, \(\mathfrak{a}\) be an ideal of \(R\) and \(M\) an \(R\)-module. \(M\) is said to be \(\mathfrak{a}\)-cofinite if Supp\((M)\subseteq V(\mathfrak{a})\) and Ext\(^i_R(R/\mathfrak{a},M)\) is finitely generated \(R\)-module for all \(i\). The paper under review is about to establish several homological theorems for \(\mathfrak{a}\)-cofinite modules which were classically stated for finitely generated ones. Among these theorems are the characterization of regular sequence in terms of the vanishing of Ext and local cohomology modules, a variant of the Auslander-Buchsbaum formula, Bass formula about the relation between injective dimension and depth, non-vanishing theorem for local cohomology modules (Corollary 3.5) and cofiniteness of some local cohomology modules (Theorem 2.1). The later two results were already stated by \textit{A. Mafi} [Arch. Math. 87, No. 3, 211--216 (2006; Zbl 1102.13018)] and by \textit{M.-T. Dibaei} and \textit{S. Yassemi} [Manuscr. Math. 117, No. 2, 199--205 (2005; Zbl 1105.13016)].