Finiteness properties of local cohomology modules for 𝔞-minimax modules

Azami, Jafar; Naghipour, Reza; Vakili, Bahram

doi:10.1090/S0002-9939-08-09530-0

Finiteness properties of local cohomology modules for $\mathfrak a$-minimax modules
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by Jafar Azami, Reza Naghipour and Bahram Vakili PDF

Proc. Amer. Math. Soc. 137 (2009), 439-448 Request permission

Abstract:

Let $R$ be a commutative Noetherian ring and $\mathfrak a$ an ideal of $R$. In this paper we introduce the concept of $\mathfrak a$-minimax $R$-modules, and it is shown that if $M$ is an $\mathfrak a$-minimax $R$-module and $t$ a non-negative integer such that $\textrm {H}_\mathfrak a^i(M)$ is $\mathfrak a$-minimax for all $i<t$, then for any $\mathfrak a$-minimax submodule $N$ of $\textrm {H}_\mathfrak a^t(M)$, the $R$-module $\textrm {Hom}_R(R/\mathfrak a,\textrm {H}_\mathfrak a^t(M)/N)$ is $\mathfrak a$-minimax. As a consequence, it follows that the Goldie dimension of $\textrm {H}_\mathfrak a^t(M)/N$ is finite, and so the associated primes of $\textrm {H}_\mathfrak a^t(M)/N$ are finite. This generalizes the main result of Brodmann and Lashgari (2000).

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