Some finiteness results for local cohomology modules with respect to a pair of ideals
Let \(M\) denote a finitely generated module over a Noetherian ring \(R\). For two ideals \(I, J\) \textit{R. Takahashi} et al. [J. Pure Appl. Algebra 213, No. 4, 582--600 (2009; Zbl 1160.13013)]) introduced the local cohomology \(H^i_{I, J}(M)\) for the pair of ideals. Note that for \(J = R\) it coincides with the ordinary local cohomology \(H^i_I(M)\). As a main result the authors prove: Suppose that \(H^i_{I, J}(M)/JH^i_{I, J}(M)\) is finitely generated for all \(i < n\) and that \(\operatorname{Tor}_1^R(R/J,L) = 0\) for any \((I,J)\)-torsion module \(L\). Then \(\operatorname{Hom}_R(R/I,H^n_{I, J}(M)/JH^n_{I, J}(M))\) is finitely generated, in particular \(\operatorname{Ass}_R H^n_{I, J}(M)/JH^n_{I, J}(M)\) is a finite set. This generalizes the classical situation for ordinary local cohomology, i.e. \(J = R\), discussed in several articles. Moreover, the authors study sufficient conditions for the finiteness of \(\operatorname{Hom}_R(R/I,H^i_{I, J}(M)/JH^i_{I, J}(M) )\).
local cohomology with respect to pair of ideals
scientific article; zbMATH DE number 7369709