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

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.

Paper structured in a new way, results unchanged. Improved exposition, some proofs altered, additional remarks added. Last section made more precise. Similar results have been independently obtained by Eardley and Manoharmayum in their ArXiv:1307.8356 preprint