Lifting Group Representations to Maximal Cohen–Macaulay Representations

AbstractAuslander announced the following result: ifRis a complete local Gorenstein ring then every finitely generatedR-module has a minimal (in the sense of4maximal Cohen–Macaulay approximation. In this paper we give a non-commutative version of Auslander's result and, in particular, show that ifRis as above and ifGis a finite group then any finitely generated representation ofGoverRhas a lifting to a representation in a maximal Cohen–Macaulay module with properties analogous to those of Auslander's approximations. WhenGis trivial, we recover Auslander's approximations. We use such a lifting to construct what we call generalized Teichmüller invariants. These will be given by a canonical embedding ofGLn(Z/(p)) intoGLm( caron p) (for somem≥n) wherepis a prime whenn=1,mwill be 1, and we get the usual Teichmüller sectionZ/(p)*→Z*p. Our proof has three ingredients. These are a version of3(see Corollaries 5.4 and 6.4), our result guaranteeing the existence of Gorenstein injective envelopes7Theorem 6.1] and a duality for non-commutative rings which generalizes Matlis duality