On the annihilators of generalized local cohomology modules

Rezaei, Shahram

doi:10.22103/jmmr.2025.24948.1774

On the annihilators of generalized local cohomology modules

Document Type : Research Paper

Author

Shahram Rezaei

Department of Mathematics, Payame Noor University, Tehran, Iran

10.22103/jmmr.2025.24948.1774

Abstract

Let ${\frak{a}}$ be an ideal of Noetherian ring $R$ and $M$, $N$ be two finitely generated $R$-modules. In this paper, we obtain some results about the annihilators of top generalized local cohomology modules. We define $T:=T_R(\frak{a},M, N)$ as the largest submodule of $N$ such that\\ $cd(\frak{a},M, T_R(\frak{a},M,N))<\operatorname{cd}(\frak{a},M,N)$. Let $(R,\frak{m})$ be a complete Gorenstein Noetherian local ring such that $pd(M)=d<\infty$, $dim\ N=dim\ R=n<\infty$ and $cd(\frak{a},M,N)=d+n$. We prove that if $Ass_R(Ext_R^d(M,N/T))\subseteq Ass_R R$, then $Ann_R(H_{\frak{m}}^{d+n}(M,N))\subseteq Ann_R Ext_R^d(M,N/T)$.

Keywords

Annihilator

local cohomology

generalized local cohomology

Subjects

Algebra

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