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

TL;DR

This paper establishes a necessary condition for characteristic zero universal deformation rings of finite group representations, showing such rings must become finite étale algebras over the fraction field, and provides examples that do not meet this criterion.

Contribution

It introduces a necessary condition linking universal deformation rings to finite étale algebras over the fraction field, and constructs explicit examples not realizable as such rings.

Findings

Universal deformation rings tensor with the fraction field form finite étale algebras.

Explicit examples of rings not obtainable as universal deformation rings.

Contrasts with prior results on profinite 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.