INVERSE PROBLEMS FOR UNIVERSAL DEFORMATION RINGS OF GROUP REPRESENTATIONS

We study representations of profinite groups over some particular type of local rings. More specifically, suppose a profinite group and its continuous finite dimensional representation over a finite field k are given. Then we are interested in studying all possibilities of lifting this representation to a representation over a ring that is complete, local, noetherian and whose residue field is isomorphic to k. For each problem of the above described type, an associated deformation functor can be defined. If such a functor is representable then the object representing it is called the universal deformation ring of the given representation. The following inverse problem is central in the thesis: which rings do occur as universal deformation rings in the introduced setting? The main results of the thesis go in two directions. Firstly, 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 representation of a profinite group. This way we completely answer the stated question in its general form. Secondly, we address its modification and provide a non-trivial necessary condition for characteristic zero universal deformation rings of representations of groups that are finite.