A uniform Artin-Rees property for syzygies in rings of dimension one and two

Let be a local Noetherian ring, let M be a finitely generated R-module and let I⊂R be an -primary ideal. Let be a free resolution of M. In this paper we study the question whether there exists an integer h such that IⁿF_i∩ker(∂_i)⊂I^n−hker(∂_i) holds for all i. We give a positive answer for rings of dimension at most two. We relate this property to the existence of an integer s such that I^s annihilates the modules for all i>0 and all integers n.