On the cofiniteness of in dimension \(<2\) local cohomology modules for a pair of ideals (Q6553567)

Let \(R\) be a commutative Noetherian ring, \(I,J\) be two ideals of \(R\), and \(M\) be an \(R\)-module. As a generaliation of the ordinary local cohomology modules with respect to an ideal, \textit{R. Takahashi} et al. [J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)] defined the local cohomology modules with respect to a pair of ideals. To be more precise, let \(\Gamma_{I,J}(M)=\{ x\in M : \exists\ t\in \mathbb N, I^t x\subseteq Jx\}\). It is easy to see that \(\Gamma_{I,J}(M)\) is a submodule of \(M\), and \(\Gamma_{I,J}(-)\) is a covariant, \(R\)-linear functor from the category of \(R\)-modules to itself. For integer \(i\), the local cohomology functor \(H^{i}_{I,J}(-)\) with respect to \((I, J)\), is defined to be the \(i\)-th right derived functor of \(\Gamma_{I,J}(-)\). Also \(H^{i}_{I,J}(M)\) is called the \(i\)-th local cohomology module of \(M\) with respect to \((I, J)\). If \(J=0\), then \(H^{i}_{I,J}(-)\) coincides with the ordinary local cohomology functor \(H^{i}_{I}(-)\).\NLet \(\mathrm{W}(I, J)=\{\mathfrak{p}\in \mathrm{Spec}(R): I^t\subseteq J+\mathfrak{p}\text{ for some positive integer }t\}\). An \(R\)-module \(M\) is called \((I,J)\)-cofinite, if \(\mathrm{Supp}_R(M)\subseteq\mathrm{W}(I,J)\), and \(\mathrm{Ext}_R^i(R/I, M)\) is finitely generated for all \(i\geq0\) [\textit{A. Tehranian} and \textit{A. Pour Eshmanan Talemi}, Bull. Iranian Math. Soc. 36, 145--155 (2010; Zbl 1223.13008)].\N\NLet \(n\) be a non-negative integer. \textit{D. Asadollahi} and \textit{R. Naghipour} [Commun. Algebra 43, No. 3, 953--958 (2015; Zbl 1318.13024)] called an \(R\)-module \(M\) in dimension \(<n\), if there is a finitely generated submodule \(N\) of \(M\) such that \(\dim\mathrm{Supp}_R(M/N) < n\).\N\NSuppose that \(t\) is a positive integer, and \(N\) is an \(R\)-module such that \(\mathrm{Ext}_R^i(R/I, N)\) is finitely generated for all \(i\leq t+1\). Also, assume that \(H^i_{I,J}(N)\) is in dimension \(<2\) for all \(i<t\).\NThe main result of the paper under review, says that the following statements are true:\N \begin{itemize}\N\item[(i)] the \(R\)-module \(H^i_{I,J}(N)\) is \((I,J)\)-cofinite for all \(i<t\);\N\item[(ii)] the \(R\)-modules \(\Hom_R(R/I, H^t_{I,J}(N))\) and \(\mathrm{Ext}_R^1(R/I, H^t_{I,J}(N))\) are finitely generated.\N\end{itemize}\N\NIt follows that, under same assumptions, the \(R\)-modules \(\Hom_R(R/I, H^t_{I,J}(N)/K)\) and \(\mathrm{Ext}_R^1(R/I, H^t_{I,J}(N)/K)\) are finitely generated, where \(K\) is a submodule of \(H^t_{I,J}(N)\) in dimension \(<1\).