Injective dimension of cofinite modules and local cohomology (Q6552643)

Let \(R\) be a commutative noetherian ring, \(\mathfrak{a}\) an ideal of \(R\), and \(M\) an \(R\)-module. For any \(i \geq 0\), the \(i\)th local cohomology module of \(M\) with respect to \(\mathfrak{a}\) is given by \N\[\NH^{i}_{\mathfrak{a}}(M) \cong \underset{n\geq 1}\varinjlim \mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a}^{n},M\right).\N\]\NHartshorne defines an \(R\)-module \(M\) to be \(\mathfrak{a}\)-cofinite if \(\mathrm{Supp}_{R}(M)\subseteq \mathrm{Var}(\mathfrak{a})\) and \(\mathrm{Ext}^{i}_{R}\left(R/ \mathfrak{a},M\right)\) is a finitely generated \(R\)-module for every \(i\geq 0\).\N\NNow suppose that \((R,\mathfrak{m})\) is local, and \(M\) is \(\mathfrak{a}\)-cofinite. In this paper, the authors show that if \(M\) has finite injective dimension, then the inequalities \N\[\N\dim(R/\mathfrak{a}) \leq \mathrm{id}_{R}(M) \leq \mathrm{depth}(\mathfrak{m},R),\N\]\Nhold, and if \(\mathfrak{m}M \neq M\), then \(\mathrm{id}_{R}(M) = \mathrm{depth}(\mathfrak{m},R)\). This generalizes the classical Bass formula for injective dimension. As an application, they obtain some results on the injective dimension of local cohomology modules. In addition, they show that \(R\) is a Cohen-Macaulay ring if it admits a Cohen-Macaulay \(R\)-module of finite projective dimension.