Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Prismatic $F$-crystals and $E$-crystalline Galois representations

Dat Pham

TL;DR

This work extends integral $p$-adic Hodge theory to coefficient fields $E$ by proving an equivalence between prismatic $F$-crystals on ${\mathcal O}_K$ (relative to $E$) and ${\mathcal O}_E$-lattices in $E$-crystalline Galois representations, generalizing the $E=\mathbf{Q}_p$ case. The authors establish a robust étale realization functor, prove full faithfulness via a Du–Liu–type descent, and obtain essential surjectivity through a transversal-prism descent argument that avoids Beilinson's fibre sequence. They connect this equivalence to Kisin–Ren’s lattice classification by specializing to Lubin–Tate prisms and to the theory of minuscule Breuil–Kisin modules via $\pi$-divisible ${\mathcal O}_E$-modules, employing Breuil–Kisin modules on the open unit disk and the Fargues–Fontaine framework. Overall, the paper provides a coherent integral classification of $E$-crystalline lattices in terms of prismatic data, with explicit links to Lubin–Tate theory and the prismatic Dieudonné correspondence. These results extend the scope of prismatic descriptions of lattices and pave the way for coefficient-ring variants of prismatic descent in $p$-adic Hodge theory.

Abstract

Let $K$ be a complete discretely valued field of mixed characteristic $(0,p)$ with perfect residue field, and let $E$ be a finite extension of $\mathbf{Q}_p$ contained in $K$. We show that the category of prismatic $F$-crystals on $\mathcal{O}_K$ (relative to $E$ in a suitable sense) is equivalent to the category of $\mathcal{O}_E$-lattices in $E$-crystalline representations defined by Kisin--Ren, extending the main result of \cite{arxiv:2106.14735} in the case $E=\mathbf{Q}_p$. As a key ingredient in the proof, by adapting a lemma of Du--Liu, we prove a general full faithfulness result for certain vector bundles on the prismatic site, which simplifies and refines the key descent step in the approach of Bhatt--Scholze without invoking the Beilinson fibre sequence.

Prismatic $F$-crystals and $E$-crystalline Galois representations

TL;DR

This work extends integral -adic Hodge theory to coefficient fields by proving an equivalence between prismatic -crystals on (relative to ) and -lattices in -crystalline Galois representations, generalizing the case. The authors establish a robust étale realization functor, prove full faithfulness via a Du–Liu–type descent, and obtain essential surjectivity through a transversal-prism descent argument that avoids Beilinson's fibre sequence. They connect this equivalence to Kisin–Ren’s lattice classification by specializing to Lubin–Tate prisms and to the theory of minuscule Breuil–Kisin modules via -divisible -modules, employing Breuil–Kisin modules on the open unit disk and the Fargues–Fontaine framework. Overall, the paper provides a coherent integral classification of -crystalline lattices in terms of prismatic data, with explicit links to Lubin–Tate theory and the prismatic Dieudonné correspondence. These results extend the scope of prismatic descriptions of lattices and pave the way for coefficient-ring variants of prismatic descent in -adic Hodge theory.

Abstract

Let be a complete discretely valued field of mixed characteristic with perfect residue field, and let be a finite extension of contained in . We show that the category of prismatic -crystals on (relative to in a suitable sense) is equivalent to the category of -lattices in -crystalline representations defined by Kisin--Ren, extending the main result of \cite{arxiv:2106.14735} in the case . As a key ingredient in the proof, by adapting a lemma of Du--Liu, we prove a general full faithfulness result for certain vector bundles on the prismatic site, which simplifies and refines the key descent step in the approach of Bhatt--Scholze without invoking the Beilinson fibre sequence.
Paper Structure (22 sections, 44 theorems, 113 equations)

This paper contains 22 sections, 44 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Sketch of the proof of Theorem 1.5
  3. E-crystalline Galois representations
  4. Definitions
  5. Relation with filtered isocrystals
  6. Theory of Breuil--Kisin modules
  7. Preliminaries
  8. Filtered isocrystals and phi-modules on the open unit disk
  9. Constructions
  10. Slopes and weak admissibility
  11. Kedlaya's slope filtration
  12. Prismatic F-crystals and E-crystalline Galois representations
  13. Preliminaries on OE-prisms
  14. Formulation of the main theorem
  15. Full faithfulness
  16. ...and 7 more sections

Key Result

Theorem 1.2

There is a commutative diagram \begin{tikzcd} \mathrm{Vect}^{\varphi}((\O_K)_{\Prism},\O_{\Prism})\ar[d,hook]\ar[r,"\simeq"] & \mathrm{Rep}_{\mathbf{Z}_p}^{\mathrm{cris}}(G_K)\ar[d,hook]\\ \mathrm{Vect}^{\varphi}((\O_K)_{\Prism},\O_{\Prism}[1/\I_{\Prism}]^{\wedge}_p)\ar[r,"\simeq"] & \mathrm

Theorems & Definitions (111)

  • Definition 1.1
  • Theorem 1.2: prismatic
  • Definition 1.4: KisinRen
  • Theorem 1.5: Theorem \ref{['main thm body']}
  • Theorem 1.6: Theorem \ref{['classification pi divisible Cheng']}
  • Proposition 1.7: Theorem \ref{['boundedness descent data']}
  • Lemma 1.8: Lemma \ref{['lemma phi(Fil) Du-Liu']}
  • Definition 2.1
  • Lemma 2.2
  • proof
  • ...and 101 more