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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
Published electronically: March 29, 2002
MathSciNet review: 1926875
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Abstract: The $i$-th local cohomology module of a finitely generated graded module $M$ over a standard positively graded commutative Noetherian ring $R$, with respect to the irrelevant ideal $R_+$, is itself graded; all its graded components are finitely generated modules over $R_0$, the component of $R$ of degree $0$. It is known that the $n$-th component $H^i_{R_+}(M)_n$ of this local cohomology module $H^i_{R_+}(M)$ is zero for all $n>> 0$. This paper is concerned with the asymptotic behaviour of $\operatorname{Ass}_{R_0}(H^i_{R_+}(M)_n)$ as $n \rightarrow -\infty$.

The smallest $i$ for which such study is interesting is the finiteness dimension $f$ of $M$ relative to $R_+$, defined as the least integer $j$ for which $H^j_{R_+}(M)$ is not finitely generated. Brodmann and Hellus have shown that $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is constant for all $n < < 0$ (that is, in their terminology, $\operatorname{Ass}_{R_0}(H^f_{R_+}(M)_n)$ is asymptotically stable for $n \rightarrow -\infty$). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that $R$ is a homomorphic image of a regular ring): our answer is precisely the set of contractions to $R_0$ of certain relevant primes of $R$ whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.

Brodmann and Hellus raised various questions about such asymptotic behaviour when $i > f$. They noted that Singh's study of a particular example (in which $f = 2$) shows that $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ need not be asymptotically stable for $n \rightarrow -\infty$. The second main aim of this paper is to determine, for Singh's example, $\operatorname{Ass}_{R_0}(H^3_{R_+}(R)_n)$ quite precisely for every integer $n$, 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

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

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

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

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