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

 

A linear function associated to asymptotic prime divisors


Authors: Daniel Katz and Eric West
Journal: Proc. Amer. Math. Soc. 132 (2004), 1589-1597
MSC (2000): Primary 13A02, 13A15, 13A30, 13E05
Published electronically: October 21, 2003
MathSciNet review: 2051118
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Abstract: Let $R$ be a Noetherian standard ${\mathbb{N}}^{\thinspace d}$-graded ring and $M,N$ finitely generated, ${\mathbb{N}}^{\thinspace d}$-graded $R$-modules. Let $I_{1}, \ldots , I_{s}$ be finitely many homogeneous ideals of $R$. We show that there exist linear functions $f,g : \mathbb{N}^{s} \to \mathbb{N}^{d}$such that the associated primes over $R_{0}$ of $[\operatorname{Ext}^{i}(N,M/I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}M)]_{m}$ and $[\operatorname{Tor}_{i}(N,M/I_{1}^{n_{1}}\cdots I_{s}^{n_{s}}M)]_{m}$ are stable whenever $m\in {\mathbb{N}}^{\thinspace d}$ satisfies $m\geq f(n_{1},\ldots ,n_{s})$ and $m\geq g(n_{1},\ldots , n_{s})$, respectively.


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Additional Information

Daniel Katz
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
Email: dlk@math.ukans.edu

Eric West
Affiliation: Department of Mathematics and Computer Science, Benedictine College, Atchison, Kansas 66002
Email: ewest@benedictine.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-07282-4
PII: S 0002-9939(03)07282-4
Keywords: Associated prime, multi-graded module, homology module
Received by editor(s): April 8, 2002
Received by editor(s) in revised form: February 13, 2003
Published electronically: October 21, 2003
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2003 American Mathematical Society