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

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.