On the associated prime ideals of local cohomology modules defined by a pair of ideals

TL;DR

This paper investigates the finiteness properties of associated prime ideals of local cohomology modules defined by a pair of ideals in a Noetherian ring, establishing conditions under which these sets are finite.

Contribution

It provides new criteria for the finiteness of associated primes of local cohomology modules with respect to a pair of ideals, extending previous results in the area.

Findings

Finiteness of associated primes of certain Ext modules under specified conditions.

Conditions ensuring the finiteness of associated primes of Hom modules involving local cohomology.

Analysis of the finiteness of associated primes for Ext modules when i=1,2.

Abstract

Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module. For a non-negative integer $n$ it is shown that, if the sets $\Ass_R(\Ext^{n} _{R}(R/I,M))$ and $\Supp_R(\Ext^{i}_{R}(R/I,H^{j}_{I,J} (M)))$ are finite for all $i \leq n+1$ and all $j< n$, then so is \linebreak$\Ass_R(\Hom_{R}(R/I,H^{n}_{I,J}(M)))$. We also study the finiteness of $\Ass_R(\Ext^{i}_{R}(R/I,H^{n}_{I,J} (M)))$ for $i=1,2$.