Lower bounds for local cohomology modules with respect to a pair of ideals (Q2809255)

Let \(I,J\) be two ideals of a commutative Noetherian ring \(R\) and \(M\) an \(R\)-module. This paper examines some finiteness properties of the modules \(H^i_{I,J}(M); \;i\in\mathbb{N}_0\).NEWLINENEWLINESet NEWLINE\[NEWLINEW(I,J)=\{\mathfrak p\in\mathrm{Spec } R|I^n\subseteq \mathfrak p+J \text{ for some positive integer } n\}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\widetilde{W}(I,J)=\{\mathfrak a|\mathfrak a\;\text{is an ideal of R and } I^n\subseteq \mathfrak a+J \text{ for some positive integer } n\}.NEWLINE\]NEWLINE The functor \(\Gamma_{I,J}\) is a subfunctor of the identity functor that determined by NEWLINE\[NEWLINE\Gamma_{I,J}(M)=\{x\in M| \mathrm{Supp}_R(Rx)\subseteq W(I,J)\}.NEWLINE\]NEWLINE For each non-negative integer \(i\), the \(i\)-th right derived functor of \(\Gamma_{I,J}\) is denoted by \(H_{I,J}^i\). Theses functors were introduced in [\textit{R. Takahashi} et al., J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)]. Note that the usual local cohomology functor \(H_{I}^i\) corresponds to the case \(J=0\).NEWLINENEWLINELet \(S\) be a Serre subcategory \(S\) of the category of \(R\)-modules that satisfies the condition \(C_I\), \(t\) a non-negative integer and \(N\) a finitely generated \(R\)-module with \(\mathrm{Supp}_RN=V(\mathfrak a)\) for some \(\mathfrak a\in \widetilde{W}(I,J)\). One of the main results of the paper says that if \(\mathrm{Ext}^j_R(N,H^i_{I,J}(M))\in S\) for all \(i<t\) and all \(j<t-i\), then \(H^i_{\mathfrak a}(M)\in S\) for all \(i<t\).NEWLINENEWLINERecall that s Serre subcategory \(S\) of the category of \(R\)-modules is said to satisfy the condition \(C_I\) if for any \(I\)-torsion \(R\)-module \(X\), \((0:_XI)\in S\) implies that \(X\in S\).