A necessary condition for characteristic zero universal deformation rings of finite group representations

Abstract

We prove the following result related to the inverse problem for universal deformation rings of group representations: Given a finite field k, denote by W(k) the ring of Witt vectors over k and by K the field of fractions of W(k). If a complete noetherian local ring R is a universal deformation ring of a representation of a finite group over k, then R \otimes_{W(k)} K is a finite \'etale K-algebra. As a consequence, we obtain explicit examples of characteristic zero complete noetherian local rings that can not be obtained as universal deformation rings of finite group representations. This contrasts with earlier results on universal deformation rings of profinite group representations.