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

Abstract

Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module of dimension $d$. For each $i\in N_0$ let $H^{i}_{I,J}(-)$ denote the $i$-th right derived functor of $\Gamma_{I,J}(-)$, where $\Gamma _{I,J}(M):=\{x \in M : I^{n}x\subseteq Jx \ \text {for} \ n\gg 1\}$. 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. Moreover, we illustrate the attached prime ideals of $H^{t}_{I,J}(M)$ on a nonlocal ring $R$, for $t= \dim M$ and $t= (I,J,M)$, where $(I,J,M)$ is the last nonvanishing level of $H^{i}_{I,J}(M)$.