Local cohomology based on a nonclosed support defined by a pair of ideals (Q1004465)

Let \(I,J\) be two ideals of a commutative noetherian ring \(R\) and let \(M\) be an \(R\)-module. In the paper under review, the authors introduced a generalization of the theory of local cohomology. They defined \((I,J)\)-torsion submodule of \(M\) by \(\Gamma_{I,J}(M):=\{x\in M:\exists n\in \mathbb{N} \text{ such that } I^nx\subseteq Jx\}\). It is easy to check that \(\Gamma_{I,J}(\cdot)\) defines a covariant additive functor from the category of \(R\)-modules and \(R\)-homomorphisms to itself which is left exact. For each non-negative integer \(i\), they defined the \(i\)-th local cohomology functor with respect to \((I,J)\), \(H_{I,J}^i(\cdot)\), to be the \(i\)-right derived functor of \(\Gamma_{I,J}(\cdot)\). Clearly when \(J=0\), then \(H_{I,J}^i(\cdot)\) coincides with the usual local cohomology functor \(H_I^i(\cdot)\). The authors developed the theory of this generalized local cohomology parallel to the existing one for usual local cohomology. In particular, they proved several vanishing results. For instance, they showed that in the case \(R\) is local \(H_{I,J}^{\dim R}(M)=0\) if and only if \(\dim \hat{R}/I\hat{R}+ \mathfrak{p}>0\) for all ideals \(\mathfrak{p}\in \text{Ass} \hat{R}\) such that \(\dim \hat{R}/\mathfrak{p} =\dim \hat{R}\) and \(J \hat{R}\subseteq \mathfrak{p}\). This can be viewed as a generalization of the Lichtenbaum-Hartshorne Vanishing Theorem. Also, they improved the Local Duality Theorem. Namely, they showed that if \(R\) is Cohen-Macaulay complete local of dimension \(d\) with the maximal ideal \(\mathfrak{m}\), \(M\) is finitely generated and \(J\) is such that \(\text{pd}_RR/J=\text{grade}(J,R)\), then \(H_{\mathfrak{m},J}^{d-i}(M)\cong \Hom_R(\text{Ext}_R^{i-t}(M,K),E_R(R/\mathfrak{m}))\) for all integers \(i\), where \(t=\text{grade}(J,R)\) and \(K=\Hom_R(H_{\mathfrak{m},J}^{d-t}(R),E_R(R/\mathfrak{m}))\).