Associated primes of graded components of local cohomology modules
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- by Markus P. Brodmann, Mordechai Katzman and Rodney Y. Sharp
- Trans. Amer. Math. Soc. 354 (2002), 4261-4283
- DOI: https://doi.org/10.1090/S0002-9947-02-02987-2
- Published electronically: March 29, 2002
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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.References
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Bibliographic Information
- Markus P. Brodmann
- Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland
- MR Author ID: 41830
- 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
- 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).
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1926875