Uniform Artin-Rees Bounds for Syzygies

TL;DR

This paper establishes a uniform bound for Artin-Rees containment in free resolutions of syzygy modules over local Noetherian rings, introducing Koszul annihilating sequences to prove the results.

Contribution

It provides the first uniform Artin-Rees bounds applicable to all ideals and resolutions of syzygy modules, advancing understanding of module structure in commutative algebra.

Findings

Uniform Artin-Rees bounds hold for all ideals and resolutions of dth syzygy modules.

Introduction of Koszul annihilating sequences as a key tool.

Bound h is independent of specific ideals and modules.

Abstract

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.