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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Published electronically: March 11, 2003
MathSciNet review: 1993206
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Abstract: It is a well-known result of M. Brodmann that if $\mathfrak{a}$ is an ideal of a commutative Noetherian ring $A$, then the set of associated primes $\operatorname{Ass} (A/\mathfrak{a}^n)$ of the $n$-th power of $\mathfrak{a}$ is constant for all large $n$. This paper is concerned with the following question: given a prime ideal $\mathfrak{p}$ of $A$ which is known to be in $\operatorname{Ass}(A/\mathfrak{a}^n)$ for all large integers $n$, can one identify a term of the sequence $(\operatorname{Ass} (A/\mathfrak{a}^n))_{n \in \mathbb{N} }$ beyond which $\mathfrak{p}$ 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.


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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: http://dx.doi.org/10.1090/S0002-9939-03-07038-2
PII: S 0002-9939(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