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A log prismatic-crystalline comparison theorem

Heng Du, Yong Suk Moon, Koji Shimizu

Abstract

We show a comparison theorem between log prismatic cohomology and log crystalline cohomology for a $p$-adic formal scheme with semistable reduction. Combined with the prismatic-étale comparison theorem recently proved by Tian, this implies the $C_{\mathrm{st}}$-conjecture in the semistable case with coefficients given by semistable local systems.

A log prismatic-crystalline comparison theorem

Abstract

We show a comparison theorem between log prismatic cohomology and log crystalline cohomology for a -adic formal scheme with semistable reduction. Combined with the prismatic-étale comparison theorem recently proved by Tian, this implies the -conjecture in the semistable case with coefficients given by semistable local systems.
Paper Structure (24 sections, 63 theorems, 200 equations)

This paper contains 24 sections, 63 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. Logarithmic prismatic sites
  3. Functoriality of log prismatic sites
  4. Prismatic F-crystals in perfect complexes
  5. Prismatic-crystalline comparison for associated sheaves
  6. Prismatic-crystalline comparison on semistable formal schemes
  7. Notation and overview
  8. Breuil--Kisin log prism
  9. Analytic prismatic $F$-crystals and perfect complexes
  10. The Breuil--Kisin and Breuil cohomologies
  11. Prismatic-crystalline comparison for crystals
  12. The Frobenius isogeny property for the crystalline and Breuil cohomology
  13. The Hyodo--Kato theory
  14. The Hyodo--Kato cohomology
  15. Application to the Cst-conjecture
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(X, M_X)$ be a proper semistable $p$-adic formal scheme over $\mathcal{O}_K$. Let $\mathbb{L}$ be a semistable $\mathbf{Z}_p$-local system on the generic fiber $X_{K}$, and let $\mathcal{E}_{\mathrm{cris}, \mathbf{Q}}$ be the corresponding filtered $F$-isocrystal on $X_0$. Then there exists an compatible with Galois actions, Frobenii, and monodromy actions, as well as filtrations after base

Theorems & Definitions (155)

  • Theorem 1.1: Theorem \ref{['thm: Cst-conj-main']}, based on Tian-prism-etale-comparison and Theorems \ref{['thm: prism-cris-comparison-BK-case-intro']}, \ref{['thm:Hyodo-Kato-complex-intro']}
  • Remark 1.2
  • Example 1.3: Breuil--Kisin log prism; Example \ref{['eg: Breuil-Kisin prism']}
  • Theorem 1.4: du-liu-moon-shimizu-purity-F-crystal; see Theorem \ref{['thm:DLMS-association-via-prismatic-F-crystal']}
  • Theorem 1.5: Theorems \ref{['thm: pris-cris-comparison-over-Breuil S']} and \ref{['thm:Frobenius-isogeny-property']}
  • Theorem 1.6: cf. Theorems \ref{['thm: pris-cris-comparison-over-Breuil S']} and \ref{['thm:Frobenius-isogeny-property']}
  • Theorem 1.7: Theorem \ref{['thm:Hyodo-Kato-cohomology']}
  • Theorem 1.8: Tian-prism-etale-comparison
  • Definition 2.1: cf. koshikawa, du-liu-moon-shimizu-purity-F-crystal
  • Definition 2.2
  • ...and 145 more