The inverse problem for universal deformation rings and the special linear group

TL;DR

This paper demonstrates that any complete noetherian local ring can be realized as a universal deformation ring of certain linear representations of profinite groups, specifically for SL_n(R) when n≥4.

Contribution

It establishes a broad realization result linking universal deformation rings to linear groups over local rings, extending known cases to higher dimensions.

Findings

Any complete noetherian local ring R is a universal deformation ring for SL_n(R) when n≥4.

The paper identifies conditions under which similar results hold for n=2 and n=3.

Provides explicit constructions of representations realizing given rings as deformation rings.

Abstract

We show that every complete noetherian local commutative ring R with residue field k can be realized as a universal deformation ring of a continuous linear representation of a profinite group. More specifically, R is the universal deformation ring of the natural representation of SL_n(R) in SL_n(k), provided that n is at least 4. We also check for which R an analogous result is true in case n=2 and n=3.