Convergence of sequences of sets of associated primes

Author:
Rodney Y. Sharp

Journal:
Proc. Amer. Math. Soc. **131** (2003), 3009-3017

MSC (2000):
Primary 13A02, 13A15, 13E05; Secondary 13A30, 13D45

DOI:
https://doi.org/10.1090/S0002-9939-03-07038-2

Published electronically:
March 11, 2003

MathSciNet review:
1993206

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Abstract | References | Similar Articles | Additional Information

Abstract: It is a well-known result of M. Brodmann that if is an ideal of a commutative Noetherian ring , then the set of associated primes of the -th power of is constant for all large . This paper is concerned with the following question: given a prime ideal of which is known to be in for all large integers , can one identify a term of the sequence beyond which will subsequently be an ever-present? This paper presents some results about convergence of sequences of sets of associated primes of graded components of finitely generated graded modules over a standard positively graded commutative Noetherian ring; those results are then applied to the above question.

**1.**M. Brodmann,*Asymptotic stability of*, Proc. Amer. Math. Soc.**74**(1979), 16-18. MR**80c:13012****2.**M. P. Brodmann and R. Y. Sharp,*Local cohomology: an algebraic introduction with geometric applications*, Cambridge University Press, 1998. MR**99h:13020****3.**W. Bruns and J. Herzog,*Cohen-Macaulay rings*, Cambridge University Press, 1993. MR**95h:13020****4.**S. McAdam and P. Eakin,*The asymptotic*Ass, J. Algebra**61**(1979), 71-81. MR**81f:13001****5.**R. Y. Sharp,*Bass numbers in the graded case, -invariant formulas, and an analogue of Faltings' Annihilator Theorem*, J. Algebra**222**(1999), 246-270. MR**2000j:13027****6.**N. V. Trung,*The Castelnuovo regularity of the Rees algebra and the associated graded ring*, Transactions Amer. Math. Soc.**350**(1998), 2813-2832. MR**98j:13006**

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Additional Information

**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/S0002-9939-03-07038-2

Keywords:
Commutative Noetherian ring,
associated prime ideal,
standard positively graded commutative Noetherian ring,
Rees ring,
Rees module,
associated graded module,
Castelnuovo regularity.

Received by editor(s):
May 10, 2002

Published electronically:
March 11, 2003

Additional Notes:
The author was partially supported by the Swiss National Foundation (Project numbers 20-52762.97 and 2000-042 129.94/1).

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2003
American Mathematical Society