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Topological Hochschild homology and integral $p$-adic Hodge theory

Bhargav Bhatt, Matthew Morrow, Peter Scholze

Abstract

In mixed characteristic and in equal characteristic $p$ we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic $K$-theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex $AΩ$ constructed in our previous work, and in equal characteristic $p$ to crystalline cohomology. Our construction of the filtration on $\mathrm{THH}$ is via flat descent to semiperfectoid rings. As one application, we refine the construction of the $AΩ$-complex by giving a cohomological construction of Breuil--Kisin modules for proper smooth formal schemes over $\mathcal O_K$, where $K$ is a discretely valued extension of $\mathbb Q_p$ with perfect residue field. As another application, we define syntomic sheaves $\mathbb Z_p(n)$ for all $n\geq 0$ on a large class of $\mathbb Z_p$-algebras, and identify them in terms of $p$-adic nearby cycles in mixed characteristic, and in terms of logarithmic de~Rham-Witt sheaves in equal characteristic $p$.

Topological Hochschild homology and integral $p$-adic Hodge theory

Abstract

In mixed characteristic and in equal characteristic we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic -theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex constructed in our previous work, and in equal characteristic to crystalline cohomology. Our construction of the filtration on is via flat descent to semiperfectoid rings. As one application, we refine the construction of the -complex by giving a cohomological construction of Breuil--Kisin modules for proper smooth formal schemes over , where is a discretely valued extension of with perfect residue field. As another application, we define syntomic sheaves for all on a large class of -algebras, and identify them in terms of -adic nearby cycles in mixed characteristic, and in terms of logarithmic de~Rham-Witt sheaves in equal characteristic .

Paper Structure

This paper contains 47 sections, 95 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. Breuil-Kisin modules
  3. Quick reminder on topological Hochschild homology (THH)
  4. From $\mathrm{THH}$ to Breuil-Kisin modules
  5. "Motivic" filtrations on $\mathrm{THH}$ and its variants
  6. Complements on cyclic homology
  7. Overview of the paper
  8. Reminders on the cotangent complex and (topological) Hochschild homology
  9. The cotangent complex
  10. Hochschild homology
  11. Topological Hochschild homology
  12. Flat descent for cotangent complexes and Hochschild homology
  13. The quasisyntomic site
  14. $p$-complete flatness
  15. The quasisyntomic site
  16. ...and 32 more sections

Key Result

Theorem 1.2

There is a $\mathfrak S$-linear cohomology theory $R\Gamma_{\mathfrak S}(\mathfrak X)$ equipped with a $\varphi$-linear Frobenius map $\varphi: R\Gamma_{\mathfrak S}(\mathfrak X)\to R\Gamma_{\mathfrak S}(\mathfrak X)$, with the following properties:

Theorems & Definitions (161)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: cf. § \ref{['sec:TCperfectoid']}
  • Definition 1.7: The quasisyntomic site, cf. Definition \ref{['defqs']}
  • Theorem 1.8: cf. Theorem \ref{['main_theorem']}
  • Remark 1.9
  • Theorem 1.10: cf. § \ref{['subsection_TC_Acrys']}
  • ...and 151 more