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•,∂•) be a free resolution of M. We find a uniform bound h such that the Artin-Rees containment InFi∩Im∂i+1⊆In−hIm∂i+1 holds for all integers i≥d, for all integers n≥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 dth syzygy modules. In order to prove our statements, we introduce the concept of Koszul annihilating sequences.