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

TL;DR

This paper investigates the structure of local cohomology modules over local rings, focusing on annihilators and attached primes, especially for relative Cohen-Macaulay modules and modules with specific cohomological properties.

Contribution

It provides new descriptions of annihilators and attached primes of local cohomology modules for relative Cohen-Macaulay modules and extends results to arbitrary modules over Noetherian rings.

Findings

Ann_R(H_{a}^{cd(a,M)}(M)) equals Ann_R M

Attached primes of H_{a}^{dim M - 1}(M) depend only on Supp(M)

For modules with cd(a,M)=cd(a,R/Ann_R M), attached primes are contained in primes with specific cohomological dimension

Abstract

Let $(R, \frak m)$ 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 $\frak a$ of $R$, then $\text{Ann}_R(H_{\mathfrak{a}}^{\text{cd}(\mathfrak{a}, M)}(M))=\text{Ann}_RM/L=\text{Ann}_RM$ and $\text{Ass}_R(R/\text{Ann}_RM)\subseteq \{\mathfrak{p} \in \text{Ass}_R M|\,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})=\text{cd}(\mathfrak{a}, M)\},$ where $L$ is the largest submodule of $M$ such that ${\rm cd}(\mathfrak{a}, L)< {\rm cd}(\mathfrak{a}, M)$. We also show that if $H^{\dim M}_{\mathfrak{a}}(M)=0$, then $\text{Att}_R(H^{\dim M-1}_{\mathfrak{a}}(M))= \{\mathfrak{p} \in \text{Supp} (M)|\,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})=\dim M-1\},$ and so the attached primes of $H^{\dim M-1}_{\mathfrak{a}}(M)$ 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 ${\rm cd}(\mathfrak{a}, M)={\rm cd}(\mathfrak{a}, R/\text{Ann}_RM)$, then $\text{Att}_R(H^{{\rm cd}(\mathfrak{a}, M)}_{\mathfrak{a}}(M))\subseteq\{\mathfrak{p} \in V(\text{Ann}_RM)|\,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})={\rm cd}(\mathfrak{a}, M)\}.$ As a consequence of this it is shown that if $\dim M=\dim R$, then $\text{Att}_R(H^{\dim M}_{\mathfrak{a}}(M))\subseteq\{\mathfrak{p} \in \text{Ass}_R M|\,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})=\dim M\}.$