Properties of local rings and resolutions of modules

We study relations between properties of different types of resolutions of modules over a commutative noetherian local ring and properties of the ring. We explain how properties of the ring impose uniform behavior on the resolutions of certain modules, and conversely, how behavior of resolutions, or of invariants, of specific modules imply nice properties of the ring.;Let R be a d-dimensional local ring containing a field, m its maximal ideal and x1,..., xd a system of parameters for. If depth R ≥ d - 1 and the local cohomology module Hd-1m R is finitely generated, then there exists an integer n such that the modules R/ xi1,...,xi d have the same Betti numbers, for all i ≥ n.;A finite R-module M is said to be Gorenstein if Exti(k,M) = 0 for all i ≠ dim R. Assume that R has a contracting endomorphism, that is to say, a homomorphism of rings ϕ: R → R such that ϕi m⊆m 2 for some i ≥ 1. Letting ϕ R denote the R-module R with action induced by ϕ, we prove: A finite R-module M is Gorenstein if and only if HomR(ϕ R,M) ≅ M and ExtiR (ϕR,M) = 0 for 1 ≤ i ≤ depth R.