A finiteness property of infinite resolutions

In this paper we prove a finiteness result for infinite minimal free resolutions over a Noetherian local ring R: If M is a module, such as the residue field, that is locally free of constant rank on the punctured spectrum of R, and I⊂R is an ideal, then the maps fn:Fn→Fn-1 in the minimal free resolution of M satisfy the uniform Artin-Rees property: INFn-1∩Imfn⊂IN-qImfn with Artin-Rees exponent q independent of n. We ask whether the same result holds for any finitely generated module, and we study some related finiteness questions.