Lifting Group Representations to Maximal Cohen–Macaulay Representations

✍ Scribed by Edgar E. Enochs; Overtoun M.G. Jenda; Jinzhong Xu

• Book ID: 102572259
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 156 KB
• Volume: 188
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.6824

✦ Synopsis

Auslander announced the following result: if R is a complete local Gorenstein Ž ring then every finitely generated R-module has a minimal in the sense of w Ž .

x. Auslander and Smalø J. Algebra 66 1980 , 61᎐122 maximal Cohen᎐Macaulay approximation. In this paper we give a non-commutative 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 representation of G over R has a lifting to a representation in a maximal Cohen᎐Macaulay module with properties analogous to those of Auslander's approximations. When G is trivial, we recover Auslander's approximations. We use such a lifting to construct what we call generalized Teichmuller Ž Ž .. invariants. These will be given by a canonical embedding of GL Zr p into n Ž . Ž .

GL Z

for some m G n where p is a prime when n s 1, m will be 1, and we m p Ž . U get the usual Teichmuller section Zr p * ª Z . Our proof has three ingredients. ¨p These are a version of Auslander and Buchweitz' result proving the existence of w Ž .