On the cofiniteness of local cohomology modules in dimension \(< 2\) (Q6116800)

Let \(R\) be a commutative Noetherian ring with identity, and let \(I\) be an ideal of \(R\). For an \(R\)-module \(M\), the \(i\)th local cohomology module of \(M\) with respect to \(I\) is defined by \[ \text{H}_{I}^i(M):=\underset{n}{\varinjlim}\ \text{Ext}_R^i(R/I^n,M). \] Many authors have investigated the finiteness properties of local cohomology modules. In particular, there are numerous results concerning the cofiniteness of these modules. Recall that an \(R\)-module \(X\) is called \(I\)-cofinite if \(\text{Supp}_R(X)\subseteq \text{V}(I)\) and the \(R\)-module \(\text{Ext}_R^i(R/I,X)\) is finitely generated for all \(i\geq 0\). Let \(n\) be a non-negative integer. An \(R\)-module \(X\) is said to be in dimension \(<n\) if there exists a finitely generated submodule \(Y\) of \(X\) such that \(\dim R/\mathfrak{p}< n\) for all \(\mathfrak{p}\in \text{Supp}_R(X/Y)\). The main result of this paper asserts the following: Theorem. Let \(t\) be a natural number. Assume that \(\text{Ext}_R^i(R/I,M)\) is finitely generated for all \(i\leq t + 1\) and \(\text{H}_{I}^i(M)\) is in dimension \(<2\) for all \(i < t\). Then \(\text{H}_{I}^i(M)\) is \(I\)-cofinite for all \(i < t\), and both \(\text{Hom}_R(R/I,\text{H}_{I}^t(M))\) and \(\text{Ext}_R^1(R/I,\text{H}_{I}^t(M))\) are finitely generated.