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Log prismatic $F$-crystals and realization functors

Kentaro Inoue

Abstract

Log prismatic cohomology theory developed by Koshikawa-Yao involves coefficient objects, called log prismatic $F$-crystals. In this paper, we construct and study realization functors from the category of log prismatic $F$-crystals to the category of coefficient objects of other $p$-adic cohomology theories, in the setting where boundary divisors may involve horizontal components.

Log prismatic $F$-crystals and realization functors

Abstract

Log prismatic cohomology theory developed by Koshikawa-Yao involves coefficient objects, called log prismatic -crystals. In this paper, we construct and study realization functors from the category of log prismatic -crystals to the category of coefficient objects of other -adic cohomology theories, in the setting where boundary divisors may involve horizontal components.
Paper Structure (22 sections, 66 theorems, 346 equations)

This paper contains 22 sections, 66 theorems, 346 equations.

Table of Contents

  1. Introduction
  2. Setups for semi-stable log formal schemes
  3. --isocrystals on log crystalline sites
  4. Log crystalline sites
  5. Isocrystals and connections
  6. --isocrystals and monodromy operators
  7. Local systems on log adic spaces
  8. Horizontal period sheaves on log adic spaces
  9. Period sheaves with connections for semi-stable log formal schemes
  10. Association for local systems on log adic spaces
  11. Log prismatic crystals
  12. Log prismatic sites
  13. "Kummer quasi-syntomic descent" for analytic log prismatic --crystals
  14. Realization functors for log prismatic crystals
  15. The étale realization functor for log prismatic --crystals
  16. ...and 7 more sections

Key Result

Theorem A

For a semi-stable log formal scheme $(\mathfrak{X},\mathcal{M}_{\mathfrak{X}})$ over $\mathcal{O}_{K}$, there exist an étale realization functor and a crystalline realization functor where $\mathrm{Loc}_{\mathbb{Z}_{p}}((\mathfrak{X},\mathcal{M}_{\mathfrak{X}})_{\eta})$ denotes the category of Kummer étale $\mathbb{Z}_{p}$-local systems on the generic fiber $(\mathfrak{X},\mathcal{M}_{\mathfrak{

Theorems & Definitions (179)

  • Definition 1.1
  • Theorem A: see Theorem \ref{['existence of filfisoc real']}
  • Remark 1.2
  • Remark 1.3
  • Theorem B: see Theorem \ref{['fully faithfulness of et real for anal pris']} and Theorem \ref{['biexactness of et real']}
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Definition 2.2: Small affine log formal schemes
  • ...and 169 more