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Convergence of sequences of sets of associated primes

Authors:	Rodney Y Sharp

Institution:	Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom

Abstract:	It is a well-known result of M. Brodmann that if $\mathfrak{a}$ is an ideal of a commutative Noetherian ring , then the set of associated primes $\operatorname{Ass} (A/\mathfrak{a}^n)$ of the -th power of $\mathfrak{a}$ is constant for all large . This paper is concerned with the following question: given a prime ideal $\mathfrak{p}$ of which is known to be in $\operatorname{Ass}(A/\mathfrak{a}^n)$ for all large integers , 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.

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

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