On the cofiniteness of local cohomology modules

Document Type : Research Paper

Author

Department of Basic Sciences, Arak University of Technology, P. O. Box 38135-1177, Arak, Iran.

10.29252/as.2021.2382

Abstract

Let $R$ be a commutative Noetherian ring with identity, $I$ be an ideal of $R$ and $M$ be an $R$-module such that $Ext^j_R(R/I, M)$ is finitely generated for all $j$. We prove that if $\dim H^i_I(M)\leq 1$ for all $i$, then for any $i \geq 0$ and for any submodule $N$ of $H^i_I(M)$ that is either $I$-cofinite or minimax, the $R$-module $H^i_I(M)/N$ is $I$-cofinite. This generalizes the main result of Bahmanpour and Naghipour [8, Theorem 2.6]. As a consequence, the Bass numbers and Betti numbers of $H^i_I (M)$ are finite for all $i \geq 0$. Also, among other things, we show that if either $\dim R/I\leq 2$ or $\dim M\leq 2$, then for each finitely generated $R$-module $N$, the $R$-module $Ext^j_R (N, H^i_I(M))$ is $I$-weakly cofinite, for all $i \geq 0$ and $j\geq 0$. This generalizes [1, Corollary 2.8].

Keywords


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