Potentially semi-stable deformations of specified Hodge-Tate type and Galois type

TL;DR

This paper proves that the set of potentially semi-stable Galois representations with fixed Hodge-Tate and Galois types forms a closed subspace within the universal deformation ring, extending previous finite field results.

Contribution

It generalizes Kisin's (2007) result to perfect fields of characteristic p > 2, establishing the closedness of certain loci in deformation rings.

Findings

Potentially semi-stable loci are closed subspaces in deformation rings.

Generalization from finite fields to perfect fields of characteristic p > 2.

Extends deformation theory results to broader field classes.

Abstract

Let $k$ be a perfect field of characteristic $p > 2$, and let $K$ be a finite totally ramified extension of $W(k)[\frac{1}{p}]$. We prove that the locus of potentially semi-stable $\mathrm{Gal}(\bar{K}/K)$-representations of a given Hodge-Tate type and Galois type is a closed subspace of the universal deformation ring, generalizing the result of Kisin (2007) where $k$ is assumed to be finite.