Equidimensionality of universal pseudodeformation rings in characteristic p for absolute Galois groups of p-adic fields

Let K be a finite extension of the p-adic field Q(p) of degree d, let F be a finite field of characteristic p and let D be an n-dimensional pseudocharacter in the sense of Chenevier of the absolute Galois group of K over the field F. For the universal mod p pseudodeformation ring R-univ (D )of D, we prove the following: The ring R(D)(ps )is equidimensional of dimension dn2 +1. Its reduced quotient R-univ (D),red contains a dense open subset of regular points x whose associated pseudocharacter D-x is absolutely irreducible and nonspecial in a certain technical sense that we shall define. Moreover, we will characterize in most cases when K does not contain a p-th root of unity the singular locus of Spec R-univ (D) . Similar results were proved by Chenevier for the generic fiber of the universal pseudodeformation ring R-univ (D) of D.