## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Cofiniteness of local cohomology modules over Noetherian local rings

#### Abstract

Let $(R,\mathfrak{m})$ be a commutative Noetherian local ring. Let $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module of dimension $d\geq 1$. In this paper we consider the $I$-cofiniteness property of the local cohomology module $H^{d-1}_I(M)$. More precisely, we prove that the $R$-module $H^{d-1}_I(M)$ is $I$-cofinite if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(M))$ is finitely generated. As an immediate consequence of this result, we prove that if $(R,\operatorname{\frak m})$ is a regular local ring of dimension $d\geq 2$ and $I$ is an ideal of $R$ with $\dim R/I\neq 1$, then $H^{d-1}_I(R)=0$ if and only if the $R$-module $\Hom_R(R/I,H^{d-1}_I(R))$ is finitely generated.

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 22, Number 5 (2015), 715-724.

Dates
First available in Project Euclid: 17 December 2015

https://projecteuclid.org/euclid.bbms/1450389243

Digital Object Identifier
doi:10.36045/bbms/1450389243

Mathematical Reviews number (MathSciNet)
MR3435077

Zentralblatt MATH identifier
1346.13032

#### Citation

Bagheriyeh, Iraj; A'zami, Jafar; Bahmanpour, Kamal. Cofiniteness of local cohomology modules over Noetherian local rings. Bull. Belg. Math. Soc. Simon Stevin 22 (2015), no. 5, 715--724. doi:10.36045/bbms/1450389243. https://projecteuclid.org/euclid.bbms/1450389243