Regular sequences and local cohomology modules with respect to a pair of ideals

Abstract

Let $R$ be a Noetherian ring, $I$ and $J$ two ideals of $R$ and $t$ an integer. Let $S$ be the class of Artinian $R$-modules, or the class of all $R$-modules $N$ with ${\dim }_{R}N\le k$, where $k$ is an integer. It is proved that $\inf\{i: H^{i}_{I,J}(M)\notin S\}=\inf\{S-\depth_\frak{a}(M): \frak{a}\in \tilde{\rm W}(I,J)\}$, where $M$ is a finitely generated $R$-module, or is a $ZD$-module such that $M/\mathfrak{a}M\notin S$ for all $\mathfrak{a}\in \tilde{\mathrm{W}}(I,J)$. Let $\Supp_R H^{i}_{I,J}(M)$ be a finite subset of $\Max(R)$ for all $i<t$. It is shown that there are maximal ideals ${\mathfrak{m}}_1,{\mathfrak{m}}_2,\dots ,{\mathfrak{m}}_k$ of $R$ such that $H_{I,J}^i(M)\cong H_{{\mathfrak{m}}_1}^i(M)\oplus H_{{\mathfrak{m}}_2}^i(M)\oplus \cdots \oplus H_{{\mathfrak{m}}_k}^i(M)$ for all $i<t$.

arXiv comment12 pages. arXiv admin note: text overlap with arXiv:1305.0164

References in corpus (1)

• Extension functors of local cohomology modules