Quasi-finite modules and asymptotic prime divisors

JOURNAL OF ALGEBRA

卷:380

Article

Katz, Daniel¹  Puthenpurakal, Tony J.² 

[1] Univ Kansas, Dept Math, Lawrence, KS 66045 USA

[2] Indian Inst Technol, Dept Math, Bombay 400076, Maharashtra, India

关键词: Quasi-finite modules;    Multigraded modules;    Asymptotic prime divisors;   

DOI  :  10.1016/j.jalgebra.2013.01.024

来源: Elsevier

【 摘 要 】

Let A be a Noetherian ring, J subset of A an ideal and C a finitely generated A-module. In this note we would like to prove the following statement. Let {I-n}(n >= 0) be a collection of ideals satisfying: (i) I-n superset of J(n), for all n, (ii) J(s) . I-s subset of Ir+s, for all r,s >= 0 and (iii) I-n subset of I-m, whenever m <= n. Then Ass(A)(InC/J(n)C) is independent of n, for n sufficiently large. Note that the set of prime ideals boolean OR((n) over dot >= 1) Ass(A)(InC/J(n)C) is finite, so the issue at hand is the realization that the primes in Ass(A)(InC/J(n)C) do not behave periodically, as one might have expected, say if circle plus(n >= 0) I-n were a Noetherian A-algebra generated in degrees greater than one. We also give a multigraded version of our results. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free