Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules

Fathi, Ali

doi:10.22055/jamm.2023.43821.2166

Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules

Document Type : Original Paper

Author

Ali Fathi

Department of Mathematics, Zanjan Branch, Islamic Azad University, Zanjan, Iran

10.22055/jamm.2023.43821.2166

Abstract

Let $R$ be a commutative Noetherian ring, and let $\mathfrak a$ be a proper ideal of $R$. Let $M$ be a non-zero finitely generated $R$-module with the finite projective dimension $p$. Also, let $N$ be a non-zero finitely generated $R$-module with $N\neq\mathfrak{a} N$, and assume that $c$ is the greatest non-negative integer with the property that $\operatorname{H}^i_{\mathfrak a}(N)$, the $i$-th local cohomology module of $N$ with respect to $\fa$, is non-zero. $\operatorname{H}^i_{\mathfrak a}(M, N)$, the $i$-th generalized local cohomology module of $M$ and $N$ with respect to $\mathfrak a$, is zero for all $i$ with $i>p+c$. In this paper, we obtain the coassociated prime ideals of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$. Using this, in the case when $R$ is a local ring and $c$ is equal to the dimension of $N$, we obtain a necessary and sufficient condition for the vanishing of $\operatorname{H}^{p+c}_{\mathfrak a}(M, N)$ which extends the Lichtenbaum-Hartshorne vanishing theorem for generalized local cohomology modules.

Highlights

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Keywords

Generalized local cohomology module

Lichtenbaum-Hartshorne vanishing theorem

coassociated prime ideal

attached prime ideal

Subjects