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Completed prismatic $F$-crystals and crystalline $\mathbf{Z}_p$-local systems

Heng Du, Tong Liu, Yong Suk Moon, Koji Shimizu

Abstract

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline étale $\mathbf{Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a complete discrete valuation ring with perfect residue field.

Completed prismatic $F$-crystals and crystalline $\mathbf{Z}_p$-local systems

Abstract

We introduce the notion of completed -crystals on the absolute prismatic site of a smooth -adic formal scheme. We define a functor from the category of completed prismatic -crystals to that of crystalline étale -local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a complete discrete valuation ring with perfect residue field.
Paper Structure (20 sections, 63 theorems, 265 equations)

This paper contains 20 sections, 63 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Review of crystalline representations and étale phi-modules
  3. Base ring
  4. Crystalline representations
  5. Étale phi-modules
  6. Completed prismatic F-crystals and crystalline representations
  7. The absolute prismatic site
  8. Completed prismatic F-crystals in the small affine case
  9. Completed prismatic F-crystals in terms of descent data
  10. Étale realization and the main theorem in the small affine case
  11. Height one case
  12. Completed prismatic F-crystals on a smooth p-adic formal scheme
  13. Quasi-Kisin modules associated with crystalline representations
  14. Quasi-Kisin modules and associated rational Kisin descent data
  15. Projectivity of M[1/p] under Assumption \ref{['assumption:base-ring-sec-3.4']}
  16. ...and 5 more sections

Key Result

Theorem 1.2

There is a contravariant equivalence of categories from the category of completed prismatic $F$-crystals on $R$ to the category of crystalline $\mathbf{Z}_p$-representations of $R[p^{-1}]$ with non-negative Hodge--Tate weights, which is functorial in $R$.

Theorems & Definitions (164)

  • Definition 1.1: (Definition \ref{['defn:category-good-prism-completed-F-crystal']}, Remark \ref{['rem:definition of completed prismatic F-crystals']})
  • Theorem 1.2: (Theorem \ref{['thm:main']})
  • Theorem 1.3: (Theorem \ref{['thm:main-global']})
  • Proposition 1.4: (Proposition \ref{['prop:equivalence-to-descent-datum']})
  • Remark 1.5
  • Definition 2.1
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Lemma 2.7
  • ...and 154 more