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

Abstract

We prove that certain crystabelline deformation rings of two dimensional residual representations of Gal(Q p /Qp) are Cohen-Macaulay. As a consequence, this allows to improve Kisin's R[1/p] = T[1/p] theorem to an R = T theorem. 2010 Mathematics Subject Classification. 11F80 and 11F85.

On local Galois deformation rings

“…Thus, 𝑅 ∞ 𝑅 𝜌 ⊗ 𝐴, where A is formally smooth over the ring ⊗ 𝑣 ∈𝑆 𝑝 \𝔭 𝑅 , 𝜉 ,𝜏 ṽ in the notation of [14]. Since the rings 𝑅 , 𝜉 ,𝜏 ṽ are O-torsion free, reduced and equi-dimensional, so is the ring A by [14,Corollary A.2] and [29,Lemma A.1]. Since 𝑅 𝜌 is also O-torsion free, reduced and equidimensional, we obtain that the same holds for 𝑅 ∞ .…”

“…It follows from [29, Lemma A.5] that after replacing L with a finite extension, we may assume that for all minimal primes of A , the quotient is geometrically integral, by which we mean that is integral domain for all finite extensions . If is a minimal prime of , then for a unique character by Corollary 4.21.…”

“…It follows from [29,Lemma A.5] that after replacing L with a finite extension, we may assume that for all minimal primes 𝔭 of A, the quotient 𝐴/𝔭 is geometrically integral, by which we mean that ( 𝐴/𝔭) ⊗ O O 𝐿 is integral domain for all finite extensions 𝐿 /𝐿. If 𝔭 is a minimal prime of 𝑅 𝜌 , then 𝑅 𝜌 /𝔭 = 𝑅 In our arguments, we will not invoke the assumption 𝑝 2𝑑, since eventually this restriction used in construction of 𝑀 ∞ should become redundant.…”