Cofiniteness and Artinianness of certain local cohomology modules (Q310479)

Let \(R\) be a Noetherian commutative ring with identity and \(I\) an ideal of \(R\). Let \(\mathcal{S}\) be a Serre subcategory of the category of \(R\)-modules and \(M\) a minimax \(R\)-module with \(\dim_RM\leq 3\). One of the main results of this paper asserts that if \(R/\mathfrak{m}\in \mathcal{S}\) for every \(\mathfrak{m}\in \mathrm{Max}(R)\), then \(\mathrm{Ext}_R^j(R/\mathfrak{m},H^i_I(M)) \in \mathcal{S}\) for every \(\mathfrak{m}\in \mathrm{Max}(R)\cap V(I)\) and all \(i,j\geq 0\). In particular, \(\mathrm{Hom}_R(R/\mathfrak{m}, H^i_I(M))\in \mathcal{S}\) for every \(\mathfrak{m}\in \mathrm{Max}(R)\cap V(I)\) and all \(i,j\geq 0\). This readily establishes the following conjecture of Huneke in the case \(\dim R\leq 3\). Conjecture: Let \((R,\mathfrak{m},k)\) be a regular local ring. Then for every ideal \(\mathfrak{a}\) of \(R\) and every nonnegative integer \(n\), the \(R\)-module \(\mathrm{Hom}_R(R/\mathfrak{m},H^n_{\mathfrak{a}}(R))\) is finitely generated. Let us recall that a full subcategory \(\mathcal{S}\) of the category of \(R\)-modules is called \textit{Serre} if it is closed under taking submodules, quotients and extensions. Also, recall that an \(R\)-module \(X\) is said to be \textit{minimax} if it has a finitely generated submodule \(Y\) such that \(R\)-module \(X/Y\) is Artinian.