# Prismatic Dieudonn\'e theory

@article{Anschutz2019PrismaticDT, title={Prismatic Dieudonn\'e theory}, author={Johannes Anschutz and Arthur-C'esar Le Bras}, journal={arXiv: Algebraic Geometry}, year={2019} }

We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $\mathrm{DF}(R)$ of \textit{filtered prismatic Dieudonn\'e crystals over $R$} and a natural functor from $p$-divisible groups over $R$ to $\mathrm{DF}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.

## 3 Citations

### Prisms and Prismatic Cohomology

- Mathematics
- 2019

We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site --- the prismatic site --- to a $p$-adic formal…

### Dieudonné theory via cohomology of classifying stacks

- MathematicsForum of Mathematics, Sigma
- 2021

Abstract We prove that if G is a finite flat group scheme of p-power rank over a perfect field of characteristic p, then the second crystalline cohomology of its classifying stack $H^2_{\text…

### Twisted differential operators and q-crystals

- Mathematics
- 2020

We describe explicitly the q-PD-envelopes considered by Bhatt and Scholze in their recent theory of q-crystalline cohomology and explain the relation with our notion of a divided polynomial twisted…

## References

SHOWING 1-10 OF 60 REFERENCES

### Prisms and Prismatic Cohomology

- Mathematics
- 2019

We introduce the notion of a prism, which may be regarded as a "deperfection" of the notion of a perfectoid ring. Using prisms, we attach a ringed site --- the prismatic site --- to a $p$-adic formal…

### Dieudonné theory over semiperfect rings and perfectoid rings

- MathematicsCompositio Mathematica
- 2018

The Dieudonné crystal of a $p$ -divisible group over a semiperfect ring $R$ can be endowed with a window structure. If $R$ satisfies a boundedness condition, this construction gives an equivalence of…

### The pro-\'etale topology for schemes

- Mathematics
- 2013

We give a new definition of the derived category of constructible $\ell$-adic sheaves on a scheme, which is as simple as the geometric intuition behind them. Moreover, we define a refined fundamental…

### Etale cohomology of diamonds

- Mathematics
- 2017

Motivated by problems on the etale cohomology of Rapoport--Zink spaces and their generalizations, as well as Fargues's geometrization conjecture for the local Langlands correspondence, we develop a…

### Divided Dieudonn\'e crystals.

- Mathematics
- 2018

We define a category of divided Dieudonn\'e crystals which classifies p-divisible groups over schemes in characteristic p with certain finiteness conditions, including all F-finite noetherian…

### Regular rings and perfect(oid) algebras

- MathematicsCommunications in Algebra
- 2019

Abstract We prove a p-adic analog of Kunz’s theorem: a p-adically complete noetherian ring is regular exactly when it admits a faithfully flat map to a perfectoid ring. This result is deduced from a…

### Topological Hochschild homology and integral p$p$-adic Hodge theory

- MathematicsPublications mathématiques de l'IHÉS
- 2019

In mixed characteristic and in equal characteristic p$p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic…

### Integral p$p$-adic Hodge theory

- MathematicsPublications mathématiques de l'IHÉS
- 2018

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of Cp$\mathbf {C}_{p}$. It takes values in a mixed-characteristic analogue of Dieudonné modules,…

### Purity for flat cohomology.

- Mathematics
- 2019

We establish the flat cohomology version of the Gabber-Thomason purity for etale cohomology: for a complete intersection Noetherian local ring $(R, \mathfrak{m})$ and a commutative, finite, flat…

### Smoothness of the truncated display functor

- Mathematics
- 2010

We show that to every p-divisible group over a p-adic ring one can associate a display by crystalline Dieudonne theory. For an appropriate notion of truncated displays, this induces a functor from…