Associated primes of graded components of local cohomology modules

Authors:
Markus P. Brodmann, Mordechai Katzman and Rodney Y. Sharp

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4261-4283

MSC (2000):
Primary 13D45, 13E05, 13A02, 13P10; Secondary 13C15

DOI:
https://doi.org/10.1090/S0002-9947-02-02987-2

Published electronically:
March 29, 2002

MathSciNet review:
1926875

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Abstract | References | Similar Articles | Additional Information

Abstract: The -th local cohomology module of a finitely generated graded module over a standard positively graded commutative Noetherian ring , with respect to the irrelevant ideal , is itself graded; all its graded components are finitely generated modules over , the component of of degree . It is known that the -th component of this local cohomology module is zero for all . This paper is concerned with the asymptotic behaviour of as .

The smallest for which such study is interesting is the finiteness dimension of relative to , defined as the least integer for which is not finitely generated. Brodmann and Hellus have shown that is constant for all (that is, in their terminology, is asymptotically stable for ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that is a homomorphic image of a regular ring): our answer is precisely the set of contractions to of certain relevant primes of whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.

Brodmann and Hellus raised various questions about such asymptotic behaviour when . They noted that Singh's study of a particular example (in which ) shows that need not be asymptotically stable for . The second main aim of this paper is to determine, for Singh's example, quite precisely for every integer , and, thereby, answer one of the questions raised by Brodmann and Hellus.

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Additional Information

**Markus P. Brodmann**

Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland

Email:
Brodmann@math.unizh.ch

**Mordechai Katzman**

Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Email:
M.Katzman@sheffield.ac.uk

**Rodney Y. Sharp**

Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Email:
R.Y.Sharp@sheffield.ac.uk

DOI:
https://doi.org/10.1090/S0002-9947-02-02987-2

Keywords:
Graded commutative Noetherian ring,
graded local cohomology module,
associated prime ideal,
ideal transform,
regular ring,
Gr\"{o}bner bases.

Received by editor(s):
November 2, 2001

Published electronically:
March 29, 2002

Additional Notes:
The third author was partially supported by the Swiss National Foundation (project number 20-52762.97).

Article copyright:
© Copyright 2002
American Mathematical Society