Prismatic Dieudonné Theory

Abstract

We define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES129 (2019), 199–310), a category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to $\mathrm {DM}^{\mathrm {adm}}(R)$ . We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.

Prismatic $G$-display and descent theory

“…If G = GL N , this result (or more precisely, the analogous result for minuscule Breuil-Kisin modules) is stated in [ALB23, Section 5.2], and the proof is given in the case where n ≤ 1. Our proof (in the case where G = GL N ) goes along the same line as that of [ALB23], but it requires some additional arguments when n ≥ 2. This result fits in the omitted part of the proof of [ALB23, Theorem 5.12]; see Section 7 for details.…”

“…The pair (S, (E)) consisting of S and the ideal (E) generated by E is a prism in the sense of Bhatt-Scholze [BS22]. In [ALB23], Anschütz-Le Bras gave a cohomological description of the functor (1.1) using the prismatic site, and proved that it is an equivalence by a different method ([ALB23, Theorem 5.12]). See also Theorem 7.2.4 in Section 7.…”

“…the ring S is noetherian, while the Zink ring W(O K ) is not). On the other hand, from the viewpoint of deformation theory, there are some technical issues in working with minuscule Breuil-Kisin modules, or prismatic Dieudonné crystals (in the sense of [ALB23]). For example, the Grothendieck-Messing deformation theory does not apply directly in this setting.…”

“…Before introducing G-µ-displays, we study Breuil-Kisin modules in Section 3, where we follow the terminology of [ALB23] and define Breuil-Kisin modules over any (A, I). We introduce the notions of displayed Breuil-Kisin module and of minuscule Breuil-Kisin module.…”

“…A φ-G-torsor over (W (R ♭ ), I R ) is also called a G-Breuil-Kisin-Fargues module for R in the literature. Let R be a complete regular local ring with residue field k. As in [BS22] and [ALB23], we consider the category (R) ∆ of bounded prisms (A, I) with a homomorphism R → A/I. A G-µ-display over (R) ∆ is defined to be an object of the following groupoid G-Disp µ ((R) ∆ ) := 2 − lim ← −(A,I)∈(R) ∆ G-Disp µ (A, I).…”