Contracting endomorphisms and Gorenstein modules

Abstract.

A finite module M over a noetherian local ring R is said to be Gorenstein if Extⁱ(k, M) = 0 for all i ≠ dim R. An endomorphism φ: R → R of rings is called contracting if $\varphi^i(\mathfrak{m}) \subseteq \mathfrak{m}^2$ for some i ≥ 1. Letting ^φR 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 (^φR, M) ≅ M and Extⁱ _R (^φR, M) = 0 for 1 ≤ i ≤ depth R.