Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A-L] W. W. Adams and P. Loustaunau, An introduction to Gröbner bases, American Mathematical Society, Providence, Rhode Island, 1994. MR 95g:13025
  • [B] M. Brodmann, A lifting result for local cohomology of graded modules, Math. Proc. Cambridge Philos. Soc. 92 (1982), 221-229. MR 84c:13014
  • [B-H] M. Brodmann and M. Hellus, Cohomological patterns of coherent sheaves over projective schemes, J. Pure and Appl. Algebra, to appear.
  • [B-M-M] M. Brodmann, C. Matteotti and Nguyen Duc Minh, Bounds for cohomological Hilbert functions of projective schemes over Artinian rings, Vietnam J. Math. 28 (2000), 341-380. MR 2001j:14022
  • [B-S] M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge University Press, 1998. MR 99h:13020
  • [G] A. Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Séminaire de Géométrie Algébrique du Bois-Marie 1962, North-Holland, Amsterdam, 1968. MR 57:16294
  • [Ma] H. Matsumura, Commutative Algebra, Benjamin, New York, 1970. MR 42:1813
  • [Mu] T. Muir, The theory of determinants in the historical order of development, Volume III, Macmillan, London, 1920
  • [S] R. Y. Sharp, Bass numbers in the graded case, $a$-invariant formulas, and an analogue of Faltings' Annihilator Theorem, J. Algebra 222 (1999), 246-270. MR 2000j:13027
  • [Si] A. K. Singh, $p$-torsion elements in local cohomology modules, Math. Research Letters 7 (2000) 165-176. MR 2001g:13039
  • [Z] V. van Zeipel, Om determinanter, hvars elementer äro binomialkoefficienter, Lunds Universitet Årsskrift ii (1865) 1-68.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 13D45, 13E05, 13A02, 13P10, 13C15

Retrieve articles in all journals with MSC (2000): 13D45, 13E05, 13A02, 13P10, 13C15


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

American Mathematical Society