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

Let (R,m,k) be a local Noetherian ring, let MM be a finitely generated RR-module and let I⊂RI⊂R be an m-primary ideal. Let F={Fi,∂i} be a free resolution of MM. In this paper we study the question whether there exists an integer hh such that InFi∩ker(∂i)⊂In−hker(∂i)InFi∩ker(∂i)⊂In−hker(∂i) holds for all ii. We give a positive answer for rings of dimension at most two. We relate this property to the existence of an integer ss such that IsIs annihilates the modules ToriR(M,R/In) for all i>0i>0 and all integers nn.