On the cofiniteness of Artinian local cohomology modules (Q2797025)

Let \((R,\mathfrak m)\) be a commutative Noetherian local ring which is a homomorphic image of a Gorenstein local ring. Let \(I\) be an ideal of \(R\), \(M\) a finitely generated \(R\)-module and \(i\) a non-negative integer. The authors show that the \(R\)-module \(H_{\mathfrak m}^i(M)\) is non-zero and \(I\)-cofinite if and only if \(\sqrt{I+\mathrm{Ann}_R (H_{\mathfrak m}^i(M))}=\mathfrak m\). Recall that an \(R\)-module \(N\) is said to be \(I\)-cofinite if \(\mathrm{Supp}_RN\subseteq V(I)\) and \(Ext_R^j(R/I,N)\) is finitely generated for all \(j\in \mathbb{N}_0\).