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

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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 SLₙ(R) in SLₙ(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.