Some results on the annihilators and attached primes of local cohomology modules

Abstract

Let $(R,\mathfrak{m})$ be a local ring and $M$ a finitely generated $R$-module. It is shown that if $M$ is relative Cohen-Macaulay with respect to an ideal $\mathfrak{a}$ of $R$, then ${\text{Ann}}_R(H_{\mathfrak{a}}^{\text{cd}(\mathfrak{a},M)}(M))={\text{Ann}}_{R}M/L={\text{Ann}}_{R}M$ and ${\text{Ass}}_R(R/{\text{Ann}}_{R}M)\subseteq \{\mathfrak{p}\in {\text{Ass}}_{R}M|\mathrm{cd}(\mathfrak{a},R/\mathfrak{p})=\text{cd}(\mathfrak{a},M)\},$ where $L$ is the largest submodule of $M$ such that $\mathrm{cd}(\mathfrak{a},L)<\mathrm{cd}(\mathfrak{a},M)$. We also show that if $H_{\mathfrak{a}}^{\dim M}(M)=0$, then ${\text{Att}}_R(H_{\mathfrak{a}}^{\dim M-1}(M))=\{\mathfrak{p}\in \text{Supp}(M)|\mathrm{cd}(\mathfrak{a},R/\mathfrak{p})=\dim M-1\},$ and so the attached primes of $H_{\mathfrak{a}}^{\dim M-1}(M)$ depends only on $\text{Supp}(M)$. Finally, we prove that if $M$ is an arbitrary module (not necessarily finitely generated) over a Noetherian ring $R$ with $\mathrm{cd}(\mathfrak{a},M)=\mathrm{cd}(\mathfrak{a},R/{\text{Ann}}_{R}M)$, then ${\text{Att}}_R(H_{\mathfrak{a}}^{\mathrm{cd}(\mathfrak{a},M)}(M))\subseteq \{\mathfrak{p}\in V({\text{Ann}}_{R}M)|\mathrm{cd}(\mathfrak{a},R/\mathfrak{p})=\mathrm{cd}(\mathfrak{a},M)\}.$ As a consequence of this it is shown that if $\dim M=\dim R$, then ${\text{Att}}_R(H_{\mathfrak{a}}^{\dim M}(M))\subseteq \{\mathfrak{p}\in {\text{Ass}}_{R}M|\mathrm{cd}(\mathfrak{a},R/\mathfrak{p})=\dim M\}.$

13 pages, to appear in Archiv der Mathematik. arXiv admin note: substantial text overlap with arXiv:1312.1253