Lifting Group Representations to Maximal Cohen–Macaulay Representations Original Research Article

Auslander announced the following result: ifRis a complete local Gorenstein ring then every finitely generatedR-module has a minimal (in the sense of[4]maximal Cohen–Macaulay approximation. In this paper we give a non-commutative version of Auslanderʹs 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 Auslanderʹs approximations. WhenGis trivial, we recover Auslanderʹs 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 of[3](see Corollaries 5.4 and 6.4), our result guaranteeing the existence of Gorenstein injective envelopes [7]Theorem 6.1] and a duality for non-commutative rings which generalizes Matlis duality.