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

TL;DR

This paper investigates the attached prime ideals of local cohomology modules defined by a pair of ideals in commutative rings, providing explicit computations using co-localization techniques for complete local rings and illustrating cases in non-local rings.

Contribution

It introduces methods to compute attached prime ideals of local cohomology modules with respect to pairs of ideals, extending understanding in both local and non-local ring contexts.

Findings

Attached prime ideals of $H^{d-1}_{I,J}(M)$ are computed via co-localization.

Explicit descriptions of attached primes for $H^{t}_{I,J}(M)$ are provided in non-local rings.

Results apply to modules over complete local rings and include cases where $t= ext{dim } M$ or $t= cd(I,J,M)$.

Abstract

Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module of dimension $d$. If $R$ is a complete local ring and $M$ is finite, then attached prime ideals of $H^{d-1}_{I,J}(M)$ are computed by means of the concept of co-localization. Also, we illustrate the attached prime ideals of $H^{t}_{I,J}(M)$ on a non-local ring $R$, for $t= \dim M$ and $t= cd(I,J,M)$.