LIFTING GROUP-REPRESENTATIONS TO MAXIMAL COHEN-MACAULAY REPRESENTATIONS

Auslander announced the following result: if R is a complete local Gor enstein ring then every finitely generated R-module has a minimal (in the sense of Auslander and Smal phi [J. Algebra 66 (1980), 61-122]) ma ximal Cohen-Macaulay approximation. In this paper we give a non-commut ative version of Auslander's result and, in particular, show that if R is as above and if G is a finite group then any finitely generated re presentation of G over R has a lifting to a representation in a maxima l Cohen-Macaulay module with properties analogous to those of Auslande r's approximations. When G is trivial, we recover Auslander's approxim ations. We use such a lifting to construct what we call generalized Te ichmuller invariants. These will be given by a canonical embedding of GL(n)(Z/(p)) into GL(m)((Z) over cap(p)) (for some m greater than or e qual to n) where p is a prime when n = 1, m will be 1, and we get the usual Teichmuller section Z/(p) --> Z(p)*. Our proof has three ingred ients. These are a version of Auslander and Buchweitz' result proving the existence of maximal Cohen-Macaulay approximations [Mem. Soc. Math . France (N.S.) 38 (1989), 5-37], (see Corollaries 5.4 and 6.4), our r esult guaranteeing the existence of Gorenstein injective envelopes [E. Enochs et al., Covers and envelopes over Gorenstein rings, Tsukuba J. Math. 20 (1996); Theorem 6.1] and a duality for non-commutative rings which generalizes Matlis duality. (C) 1997 Academic Press.