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LOCAL-GLOBAL PRINCIPLE FOR THE FINITENESS AND ARTINIANNESS OF GENERALISED LOCAL COHOMOLOGY MODULES

Part of: Chain conditions, finiteness conditions Commutative algebra: Homological methods

Published online by Cambridge University Press:  20 August 2012

ALI FATHI
ALI FATHI*
Affiliation:
Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran (email: [email protected])
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Abstract

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Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules, where $R$ is a commutative Noetherian ring. Let $\mathfrak a$ and $\mathfrak b$ be ideals of $R$ and let $M$ and $N$ be finite $R$-modules. We prove that if $N$ and $H^i_{\mathfrak a}(M,N)$ belong to $\mathcal S$ for all $i\lt n$ and if $n\leq \mathrm {f}$-$\mathrm {grad}({\mathfrak a},{\mathfrak b},N )$, then $\mathrm {Hom}_{R}(R/{\mathfrak b},H^n_{{\mathfrak a}}(M,N))\in \mathcal S$. We deduce that if either $H^i_{\mathfrak a}(M,N)$ is finite or $\mathrm {Supp}\,H^i_{\mathfrak a}(M,N)$ is finite for all $i\lt n$, then $\mathrm {Ass}\,H^n_{\mathfrak a}(M,N)$ is finite. Next we give an affirmative answer, in certain cases, to the following question. If, for each prime ideal ${\mathfrak {p}}$ of $R$, there exists an integer $n_{\mathfrak {p}}$ such that $\mathfrak b^{n_{\mathfrak {p}}} H^i_{\mathfrak a R_{\mathfrak {p}}}({M_{\mathfrak {p}}},{N_{\mathfrak {p}}})=0$ for every $i$ less than a fixed integer $t$, then does there exist an integer $n$ such that $\mathfrak b^nH^i_{\mathfrak a}(M,N)=0$ for all $i\lt t$? A formulation of this question is referred to as the local-global principle for the annihilation of generalised local cohomology modules. Finally, we prove that there are local-global principles for the finiteness and Artinianness of generalised local cohomology modules.

Keywords

(generalised) local cohomology modulefinitenessArtiniannesslocal-global principlefilter regular sequence

MSC classification

Secondary: 13D45: Local cohomology 13E10: Artinian rings and modules, finite-dimensional algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 3 , June 2013 , pp. 480 - 492
Copyright
Copyright © 2012 Australian Mathematical Publishing Association Inc. 

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