On the attached primes of top local cohomology modules

Abstract

Let $\mathfrak{a}$ be an ideal of Noetherian ring R and M a finitely generated R-module such that $\text{cd}(\mathfrak{a},M)=c$. In this paper, we investigate ${\text{Att}}_R({\mathrm{H}}_{\mathfrak{a}}^c(M))$. Among other things, it is shown that $\mathrm{Max}\{\mathfrak{p}\in {\text{Supp}}_{R}M|\text{cd}(\mathfrak{a},R/\mathfrak{p})=c\}\subseteq {\text{Att}}_R({\mathrm{H}}_{\mathfrak{a}}^c(M))$. We also show that ${\text{Att}}_R({\mathrm{H}}_{\mathfrak{a}}^c(M))=\{\mathfrak{p}\in {\text{Supp}}_{R}M|\text{cd}(\mathfrak{a},R/\mathfrak{p})=c, \mathfrak{p}={\text{Ann}}_R({\mathrm{H}}_{\mathfrak{a}}^c(R/\mathfrak{p}))\}$ and $\begin{array}{c}\{\mathfrak{p}\in {\text{Supp}}_{R}M|\text{cd}(\mathfrak{a},R/\mathfrak{p})=c,\text{dim}R/\mathfrak{p}-1\le \text{cd}(\mathfrak{a},R/\mathfrak{p})\le \text{dim}R/\mathfrak{p}\}\\ \subseteq {\text{Att}}_R({\mathrm{H}}_{\mathfrak{a}}^c(M)).\end{array}$ Finally, we prove that if $(R,\mathfrak{m})$ is a local ring and $\text{dim}R/\mathfrak{a}=1$ then ${\text{Att}}_R({\mathrm{H}}_{\mathfrak{a}}^c(M))=\{\mathfrak{p}\in {\text{Supp}}_{R}M|\text{cd}(\mathfrak{a},R/\mathfrak{p})=\text{cd}(\mathfrak{a},M)\}$. Then by using this, it is shown that if $(R,\mathfrak{m})$ is a local ring then $\{\mathfrak{p}\in {\text{Supp}}_{R}M|\text{cd}(\mathfrak{a},R/\mathfrak{p})=c,\text{dim}R/(\mathfrak{a}+\mathfrak{p})=1\}\subseteq {\text{Att}}_R({\mathrm{H}}_{\mathfrak{a}}^c(M)).$

KEYWORDS:

• Attached primes
• local cohomology

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D45
• 14B15
• 13E05

Acknowledgments

The author would like to thank the referee for a careful reading and many useful suggestions.