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On the dimension and multiplicity of local cohomology modules

Published online by Cambridge University Press:  22 January 2016

Markus P. Brodmann  and
Rodney Y. Sharp
Markus P. Brodmann
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 8057 Zürich, Switzerland, Brodmann@math.unizh.ch
Rodney Y. Sharp
Affiliation:
Department of Pure Mathematics University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom, R.Y.Sharp@sheffield.ac.uk
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Abstract

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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 pseudosupports 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.

Keywords

13C1513D4513H1513D4013E1013F4013H1013J10
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 167 , 2002 , pp. 217 - 233
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

References

[1] Brodmann, M., A particular class of regular domains, J. Algebra, 54 (1978), 366–373.Google Scholar
[2] Brodmann, M. and Rotthaus, C., Local domains with bad sets of formal prime divisors, J. Algebra, 75 (1982), 386–394.Google Scholar
[3] Brodmann, M. P. and Sharp, R. Y., Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, 1998.Google Scholar
[4] Bruns, W. and Herzog, J., Cohen–Macaulay rings, Cambridge University Press, 1993.Google Scholar
[5] Greco, S., Two theorems on excellent rings, Nagoya Math. J., 60 (1976), 139–149.Google Scholar
[6] Hutchins, H. C., Examples of commutative rings, Polygonal, Passaic, New Jersey, 1981.Google Scholar
[7] Kirby, D., Artinian modules and Hilbert polynomials, Quart. J. Math. Oxford (2), 24 (1973), 47–57.Google Scholar
[8] Matsumura, H., Commutative algebra, Benjamin, New York, 1970.Google Scholar
[9] Matsumura, H., Commutative ring theory, Cambridge University Press, 1986.Google Scholar
[10] Melkersson, L. and Schenzel, P., The co-localization of an Artinian module, Proc. Edinburgh Math. Soc., 38 (1995), 121–131.CrossRefGoogle Scholar
[11] Roberts, R. N., Krull dimension for Artinian modules over quasi local commutative rings, Quart. J. Math. Oxford (2), 26 (1975), 269–273.Google Scholar
[12] Sharp, R. Y., Some results on the vanishing of local cohomology modules, Proc. London Math. Soc. (3), 30 (1975), 177–195.Google Scholar