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On the dimension and multiplicity of local cohomology modules


Brodmann, M; Sharp, R (2002). On the dimension and multiplicity of local cohomology modules. Nagoya Mathematical Journal, 167:217-233.

Abstract

This paper is concerned with a finitely generated module $M$ over a(commutative Noetherian) local ring $R$. In the case when $R$ is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of $M$ with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when $R$ is universally catenary and such that all its formal fibres are Cohen-Macaulay. These formulae involve certain subsets of the spectrum of $R$ called the pseudo-supports of $M$; these pseudo-supports are closed in the Zariski topology when $R$ is universally catenary and has the property that all its formal fibres are Cohen-Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

This paper is concerned with a finitely generated module $M$ over a(commutative Noetherian) local ring $R$. In the case when $R$ is a homomorphic image of a Gorenstein local ring, one can use the well-known associativity formula for multiplicities, together with local duality and Matlis duality, to produce analogous associativity formulae for the local cohomology modules of $M$ with respect to the maximal ideal. The main purpose of this paper is to show that these formulae also hold in the case when $R$ is universally catenary and such that all its formal fibres are Cohen-Macaulay. These formulae involve certain subsets of the spectrum of $R$ called the pseudo-supports of $M$; these pseudo-supports are closed in the Zariski topology when $R$ is universally catenary and has the property that all its formal fibres are Cohen-Macaulay. However, examples are provided to show that, in general, these pseudo-supports need not be closed. We are able to conclude that the above-mentioned associativity formulae for local cohomology modules do not hold over all local rings.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Uncontrolled Keywords:Artinian module; multiplicity of local cohomology module; Cohen-Macaulay fibers; universally catenary module; Matlis dual; Noetherian local ring
Language:English
Date:2002
Deposited On:27 May 2010 16:08
Last Modified:05 Apr 2016 13:25
Publisher:Nagoya Daigaku
ISSN:0027-7630
Official URL:http://www.math.nagoya-u.ac.jp/en/journal/data/2002.html#167
Related URLs:http://projecteuclid.org/euclid.nmj/1114649297
http://www.zentralblatt-math.org/zmath/en/search/?q=an:1044.13007
http://www.ams.org/mathscinet-getitem?mr=1924724
Permanent URL: https://doi.org/10.5167/uzh-21933

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