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

Let image be a local Noetherian ring, let M be a finitely generated R-module and let Isubset ofR be an image-primary ideal. Let image be a free resolution of M. In this paper we study the question whether there exists an integer h such that InFi∩ker(∂i)subset ofIn−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 Is annihilates the modules image for all i>0 and all integers n.