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

TL;DR

This paper investigates the properties of local cohomology modules with respect to a pair of ideals in a Noetherian ring, establishing relationships between their depths, supports, and decompositions, with implications for Artinian and dimension-bounded modules.

Contribution

It provides new characterizations of the infimum of indices where local cohomology modules exit certain classes, and demonstrates their decomposition into modules supported at finitely many maximal ideals.

Findings

Characterization of the infimum index for local cohomology modules outside a class S.

Decomposition of local cohomology modules into direct sums supported at finitely many maximal ideals.

Conditions under which the support of local cohomology modules is finite.

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 $\dim_RN\leq k$, 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 $M/\frak{a}M\notin S$ for all $\frak{a}\in \tilde{\rm W}(I,J)$. 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 $\frak m_1, \frak m_2,\ldots,\frak m_k$ of $R$ such that $H^{i}_{I,J}(M)\cong H^{i}_{\frak m_1}(M)\oplus H^{i}_{\frak m_2}(M)\oplus\cdots\oplus H^{i}_{\frak m_k}(M)$ for all $i<t$.