Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Syntomification and crystalline local systems

Dylan Pentland

TL;DR

The paper establishes a bridge between syntomic geometry and p-adic Hodge theory by showing that reflexive sheaves on the syntomic stack $\mathrm{X}^{\Syn}$ correspond to $\mathbf{Z}_p$-lattices in crystalline local systems on the rigid generic fiber $\mathrm{X}_{\eta}$, and uses this to analyze the essential image of the étale realization functor on the isogeny category of perfect complexes on $\mathrm{X}^{\Syn}$. It leverages Breuil–Kisin module theory, prismatic $F$-crystals, and the $T_{\mathrm{ét}}$ equivalence to relate crystalline Galois representations with derived objects, via a pushout construction of $\mathrm{X}^{\Syn}$ from formal stacks. In the smooth and proper case, $\mathsf{Perf}(\mathrm{X}^{\Syn})[1/p]$ is shown to be equivalent to the derived category of admissible filtered $F$-isocrystals in perfect complexes, equivalently $\D^{b}(\Rep^{\cris}_{\Q_p}(\mathrm{G}_K))$ and $\D^{b}(\mathrm{MF}_{K}^{\varphi,\mathrm{wa}})$, following Colmez–Fontaine's equivalence. The approach uses a derived Beilinson fiber square to formulate a coherent notion of derived filtered $F$-isocrystals and proves cohomological comparisons between syntomic and étale realizations, with the syntomic period map becoming injective on cohomology in the proper setting.

Abstract

Let $p$ be a prime, and let $\mathrm{X}$ be a smooth $p$-adic formal scheme over $\mathrm{Spf} \mathcal{O}_K$ where $K/\mathbf{Q}_p$ is a finite extension. We show that reflexive sheaves on the stack $\mathrm{X}^{\mathrm{Syn}}$ are equivalent to $\mathbf{Z}_p$-lattices in crystalline local systems on the rigid generic fiber $\mathrm{X}_η$, and then use this to study the essential image of the étale realization functor on the isogeny category of perfect complexes on $\mathrm{X}^{\mathrm{Syn}}$. We also show that when $\mathrm{X}/\mathrm{Spf} \mathcal{O}_K$ is smooth and proper that $\mathsf{Perf}(\mathrm{X}^{\mathrm{Syn}})[1/p]$ is equivalent to a category of admissible filtered $F$-isocrystals in perfect complexes.

Syntomification and crystalline local systems

TL;DR

The paper establishes a bridge between syntomic geometry and p-adic Hodge theory by showing that reflexive sheaves on the syntomic stack correspond to -lattices in crystalline local systems on the rigid generic fiber , and uses this to analyze the essential image of the étale realization functor on the isogeny category of perfect complexes on . It leverages Breuil–Kisin module theory, prismatic -crystals, and the equivalence to relate crystalline Galois representations with derived objects, via a pushout construction of from formal stacks. In the smooth and proper case, is shown to be equivalent to the derived category of admissible filtered -isocrystals in perfect complexes, equivalently and , following Colmez–Fontaine's equivalence. The approach uses a derived Beilinson fiber square to formulate a coherent notion of derived filtered -isocrystals and proves cohomological comparisons between syntomic and étale realizations, with the syntomic period map becoming injective on cohomology in the proper setting.

Abstract

Let be a prime, and let be a smooth -adic formal scheme over where is a finite extension. We show that reflexive sheaves on the stack are equivalent to -lattices in crystalline local systems on the rigid generic fiber , and then use this to study the essential image of the étale realization functor on the isogeny category of perfect complexes on . We also show that when is smooth and proper that is equivalent to a category of admissible filtered -isocrystals in perfect complexes.
Paper Structure (1 section, 2 theorems, 11 equations)

This paper contains 1 section, 2 theorems, 11 equations.

Table of Contents

  1. Introduction

Key Result

Theorem 1

Let $K/\Q_p$ be a finite extension. Then there is a symmetric monoidal equivalence Moreover, the diagram

Theorems & Definitions (5)

  • Theorem 1: Bhatt-Scholze, prismFcris
  • proof : Proof of Proposition \ref{['prop:O_K_cohom']}
  • Remark 2
  • Proposition 3
  • proof