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A remark on crystalline cohomology

Moritz Kerz, Georg Tamme

TL;DR

The paper presents a higher-categorical construction of crystalline cohomology by lifting smooth $k$-algebras to $p$-adic complete smooth $R$-algebras and organizing lifts up to coherent homotopy. It builds a simplicially enriched framework of formal schemes and a PD-de Rham theory, culminating in a functor CrIs: $\mathrm{Sm}_{\overline R}^{\mathrm{op}}\to \mathrm{D}(R)^{\wedge}$ that defines crystalline cohomology with descent properties. A simplicial PD-de Rham object $\mathrm{dR}^{\Delta}$ is introduced, and crystalline cohomology is shown to satisfy étale descent and extend via Yoneda to sheaves, with a precise comparison to Berthelot’s classical crystalline theory. The main contributions are the higher-categorical formulation of liftings, the simplicial enrichment of formal schemes, and the demonstration that the new crystalline cohomology agrees with the traditional de Rham theory on liftings and with Berthelot’s theory via a robust comparison.

Abstract

We propose a new approach to crystalline cohomology based on the observation that one can lift smooth algebras uniquely "up to coherent homotopy."

A remark on crystalline cohomology

TL;DR

The paper presents a higher-categorical construction of crystalline cohomology by lifting smooth -algebras to -adic complete smooth -algebras and organizing lifts up to coherent homotopy. It builds a simplicially enriched framework of formal schemes and a PD-de Rham theory, culminating in a functor CrIs: that defines crystalline cohomology with descent properties. A simplicial PD-de Rham object is introduced, and crystalline cohomology is shown to satisfy étale descent and extend via Yoneda to sheaves, with a precise comparison to Berthelot’s classical crystalline theory. The main contributions are the higher-categorical formulation of liftings, the simplicial enrichment of formal schemes, and the demonstration that the new crystalline cohomology agrees with the traditional de Rham theory on liftings and with Berthelot’s theory via a robust comparison.

Abstract

We propose a new approach to crystalline cohomology based on the observation that one can lift smooth algebras uniquely "up to coherent homotopy."
Paper Structure (7 sections, 11 theorems, 33 equations)

This paper contains 7 sections, 11 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. A simplicial ring
  3. Lifting smooth algebras
  4. Simplicially enriched formal schemes
  5. PD-de Rham cohomology
  6. A new approach to crystalline cohomology
  7. Comparison

Key Result

Lemma 2.1

The canonical map $R\Delta_\bullet \xrightarrow{\sim} R$ induced by $T_i \mapsto 0$ is a trivial Kan fibration.

Theorems & Definitions (28)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Claim 2.4
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 18 more