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,..., x_d a system of parameters for. If depth R ≥ d - 1 and the local cohomology module [special characters omitted] is finitely generated, then there exists an integer n such that the modules R/[special characters omitted] 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[special characters omitted] 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 Hom_R(^ϕ R,M) ≅ M and [special characters omitted](^ϕR,M) = 0 for 1 ≤ i ≤ depth R.