Uniform Artin-Rees Bounds for Syzygies

Let $(R,m)$ be a local Noetherian ring, let $M$ be a finitely generated $R$-module and let $(F_{\bullet},\partial_{\bullet})$ be a free resolution of $M$. We find a uniform bound $h$ such that the Artin-Rees containment $I^n F_i\cap Im \, \partial_{i+1} \subseteq I^{n-h} Im \, \partial_{i+1}$ holds...

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Bibliographic Details
Main Authors Aberbach, Ian M, Hosry, Aline, Striuli, Janet
Format Journal Article
LanguageEnglish
Published 11.06.2014
Subjects
Mathematics - Commutative Algebra
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Summary:Let $(R,m)$ be a local Noetherian ring, let $M$ be a finitely generated $R$-module and let $(F_{\bullet},\partial_{\bullet})$ be a free resolution of $M$. We find a uniform bound $h$ such that the Artin-Rees containment $I^n F_i\cap Im \, \partial_{i+1} \subseteq I^{n-h} Im \, \partial_{i+1}$ holds for all integers $i\ge d$, for all integers $n\ge h$, and for all ideals $I$ of $R$. In fact, we show that a considerably stronger statement holds. The uniform bound $h$ holds for all ideals and all resolutions of $d$th syzygy modules. In order to prove our statements, we introduce the concept of Koszul annihilating sequences.
DOI:10.48550/arxiv.1406.2866