Cofiniteness of local cohomology modules over Noetherian local rings (Q907812)

Let \((R,\mathfrak m)\) be a commutative Noetherian local ring with identity. Let \(I\) be an ideal of \(R\) and \(M\) an \(R\)-module. The paper under review deals with \(I\)-cofiniteness of the local cohomology modules \[ \mathrm{H}_{I}^i(M):=\underset{\ell\in \mathbb{N}_0} {\varinjlim}\mathrm{Ext}^i_R(R/{I}^{\ell},M); \;i\in \mathbb{N}_0. \] Recall that an \(R\)-module \(X\) is said to be \(I\)-cofinite if \(\mathrm{Supp}_RX\subseteq \mathrm{V}(I)\) and the \(R\)-module \(\mathrm{Ext}^i_R(R/{I},X)\) is finitely generated for every \(i\in \mathbb{N}_0\). Assume that \(M\) is finitely generated and \(d:=\dim_RM\geq 1\). The main result of the paper asserts that \(\mathrm{H}_{I}^{d-1}(M)\) is \(I\)-cofinite if and only if \(\mathrm{Hom}_R(R/I,\mathrm{H}_{I}^{d-1}(M))\) is finitely generated. Next, assume that \(R\) is regular of dimension \(n\geq 2\) and \(\dim R/I\geq 2\). As an application, the authors deduce that \(\mathrm{H}_{I}^{n-1}(R)=0\) if and only if \(\mathrm{Hom}_R(R/I,\mathrm{H}_{I}^{n-1}(R))\) is finitely generated.