Uniform Artin-Rees Bounds for Syzygies

Let $(R,m)$ be a local Noetherian ring, let $M$ be a finitely generated $R$-module and let $(F_{\bullet },{\mathrm{\partial }}_{\bullet })$ 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.