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A linear function associated to asymptotic prime divisors

Authors:	Daniel Katz Eric West

Institution:	Department of Mathematics, University of Kansas, Lawrence, Kansas 66045 ; Department of Mathematics and Computer Science, Benedictine College, Atchison, Kansas 66002

Abstract:	Let be a Noetherian standard ${\mathbb{N}}^{\thinspace d}$ -graded ring and finitely generated, ${\mathbb{N}}^{\thinspace d}$ -graded -modules. Let $I_{1}, \ldots , I_{s}$ be finitely many homogeneous ideals of . 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.

Keywords:	Associated prime multi-graded module homology module

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