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Log prismatic $F$-crystals and purity

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

Abstract

Our goal is to study $p$-adic local systems on a rigid-analytic variety with semistable formal model. We prove that such a local system is semistable if and only if so are its restrictions to the points corresponding to the irreducible components of the special fiber. For this, the main body of the paper concerns analytic prismatic $F$-crystals on the absolute logarithmic prismatic site of a semistable $p$-adic log formal scheme. Analyzing Breuil-Kisin log prisms, we obtain a prismatic purity theorem and deduce the above purity theorem for semistable local systems.

Log prismatic $F$-crystals and purity

Abstract

Our goal is to study -adic local systems on a rigid-analytic variety with semistable formal model. We prove that such a local system is semistable if and only if so are its restrictions to the points corresponding to the irreducible components of the special fiber. For this, the main body of the paper concerns analytic prismatic -crystals on the absolute logarithmic prismatic site of a semistable -adic log formal scheme. Analyzing Breuil-Kisin log prisms, we obtain a prismatic purity theorem and deduce the above purity theorem for semistable local systems.
Paper Structure (25 sections, 86 theorems, 214 equations)

This paper contains 25 sections, 86 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Absolute logarithmic prismatic site
  3. Definition of the absolute logarithmic prismatic site
  4. Breuil--Kisin log prism and its self-products
  5. Breuil--Kisin log prism in the CDVR case
  6. Ring-theoretic techniques for purity
  7. Analytic prismatic F-crystals, étale and crystalline realizations
  8. Analytic prismatic F-crystals
  9. Analytic prismatic F-crystals in the small affine case
  10. Laurent F-crystals and étale realizations
  11. Semistable Zp-local systems
  12. Crystalline realization
  13. Association of étale and crystalline realizations
  14. Semistable representations in the CDVR case
  15. Classically semistable representations
  16. ...and 10 more sections

Key Result

Theorem 1.1

A $p$-adic local system on $\mathcal{X}$ is semistable if and only if its restriction to each $X$-Shilov point corresponds to a semistable Galois representation.

Theorems & Definitions (212)

  • Theorem 1.1: Purity for semistable local systems
  • Example 1.2: Breuil--Kisin log prism
  • Theorem 1.3: bhatt-scholze-prismaticFcrystaldu-liu-prismaticphiGhatmodule
  • Theorem 1.4: GuoReinecke-Ccris, du-liu-moon-shimizu-completed-prismatic-F-crystal-loc-system
  • Theorem 1.5: Theorem \ref{['thm:logetalerealizationofLaurentFcrystals-strict']}, Remark \ref{['rem:strict vs saturated for Laurent F crystals']}
  • Theorem 1.6: Purity for analytic prismatic $F$-crystals: Theorem \ref{['thm:main-purity']}
  • Theorem 1.7: Part of Theorem \ref{['thm:CDVR-semistable-notions-equivalent']}
  • Theorem 1.8: Corollary \ref{['cor:semistable-prismatic-Faltings-equivalent']}
  • Lemma 2.1
  • proof
  • ...and 202 more