On crystabelline deformation rings of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ (with an appendix by Jack Shotton)

TL;DR

This paper proves that certain crystabelline deformation rings of 2-dimensional Galois representations are Cohen-Macaulay, enabling an enhancement of Kisin's theorem from an equality after inverting p to an exact equality.

Contribution

It establishes the Cohen-Macaulay property for specific crystabelline deformation rings, leading to a stronger form of Kisin's R=T theorem.

Findings

Cohen-Macaulay property proven for certain deformation rings

Improved Kisin's R=T theorem to R= T without inverting p

Enhanced understanding of the structure of Galois deformation rings

Abstract

We prove that certain crystabelline deformation rings of two dimensional residual representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$ are Cohen-Macaulay. As a consequence, this allows to improve Kisin's $R[1/p]=\mathbb{T}[1/p]$ theorem to an $R=\mathbb{T}$ theorem.