On local Galois deformation rings

Abstract

We show that framed deformation rings of mod $p$ representations of the absolute Galois group of a $p$-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application we prove density results of loci with prescribed $p$-adic Hodge theoretic properties.Comment: Revised version after a referee report. Lemma 3.21 has a better proof following a suggestion of the referee, and the appendix on Kummer-irreducible points has been rewritte