Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Absolute prismatic cohomology

Bhargav Bhatt, Jacob Lurie

TL;DR

The paper develops a geometric framework for absolute prismatic cohomology by associating prismatic crystals on Spf(Z_p) with quasi-coherent sheaves on the Cartier-Witt stack $ ext{WCart}$. It introduces the Nygaard filtration, the diffracted Hodge complex, and the Hodge-Tate divisor to link prismatic data with de Rham and crystalline theories, and defines syntomic cohomology with canonical Chern classes to connect to étale and K-theoretic invariants. Central innovations include the Breuil-Kisin twist, the prismatic logarithm, and a robust stacky calculus on $ ext{WCart}$ and its HT locus, enabling explicit comparisons and descent results across absolute and relative prismatic cohomology. The framework yields practical computational tools (e.g., using the q-de Rham prism) and clarifies how Frobenius and the Sen operator govern the interaction between HT and non-HT components, with far-reaching implications for $p$-adic Hodge theory and related cohomological theories.

Abstract

The goal of this paper is to study the absolute prismatic cohomology of $p$-adic formal schemes. We do so by recasting the notion of a prismatic crystal on $\mathrm{Spf}(\mathbf{Z}_p)$ in terms of quasicoherent sheaves on a geometric object we call the Cartier-Witt stack.

Absolute prismatic cohomology

TL;DR

The paper develops a geometric framework for absolute prismatic cohomology by associating prismatic crystals on Spf(Z_p) with quasi-coherent sheaves on the Cartier-Witt stack . It introduces the Nygaard filtration, the diffracted Hodge complex, and the Hodge-Tate divisor to link prismatic data with de Rham and crystalline theories, and defines syntomic cohomology with canonical Chern classes to connect to étale and K-theoretic invariants. Central innovations include the Breuil-Kisin twist, the prismatic logarithm, and a robust stacky calculus on and its HT locus, enabling explicit comparisons and descent results across absolute and relative prismatic cohomology. The framework yields practical computational tools (e.g., using the q-de Rham prism) and clarifies how Frobenius and the Sen operator govern the interaction between HT and non-HT components, with far-reaching implications for -adic Hodge theory and related cohomological theories.

Abstract

The goal of this paper is to study the absolute prismatic cohomology of -adic formal schemes. We do so by recasting the notion of a prismatic crystal on in terms of quasicoherent sheaves on a geometric object we call the Cartier-Witt stack.
Paper Structure (77 sections, 245 theorems, 925 equations)

This paper contains 77 sections, 245 theorems, 925 equations.

Table of Contents

  1. Introduction
  2. Goal
  3. Some recent history
  4. Absolute prismatic cohomology and the Cartier-Witt stack
  5. Diffracted Hodge cohomology and the Hodge-Tate divisor
  6. The Nygaard filtration
  7. Syntomic cohomology
  8. Cochains and Animation
  9. The Bestiary
  10. Completions
  11. Acknowledgments
  12. Breuil-Kisin Twists and the Prismatic Logarithm
  13. Transversal Prisms
  14. Breuil-Kisin Twists: Transversal Case
  15. The Prismatic Logarithm: Transversal Case
  16. ...and 62 more sections

Key Result

Theorem 1.2.1

Let $K$ be an algebraically closed field which is complete with respect to a $p$-adic absolute value. To every smooth and proper $\mathop{\mathrm{\mathcal{O}}}\nolimits_{K}$-scheme $X$, one can associate perfect complex of $A_{\mathrm{inf}}$-modules $\mathop{\mathrm{R \Gamma}}\nolimits_{ A_{\mathrm{

Theorems & Definitions (817)

  • Theorem 1.2.1: BMS1
  • Definition 1.2.2: Prisms: Torsion-Free Case
  • Example 1.2.4
  • Example 1.2.5: Crystalline cohomology
  • Example 1.2.6: $q$-de Rham cohomology
  • Example 1.3.1: The prismatic cohomology sheaf
  • Example 1.3.2: The Breuil-Kisin twist
  • Example 1.3.3: $\mathop{\mathrm{WCart}}\nolimits$ via simple quotient stacks
  • Remark 1.3.4: Prismatic $F$-crystals
  • Remark 1.3.5: Drinfeld's stacks
  • ...and 807 more