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Integral p-adic Hodge theory

Bhargav Bhatt, Matthew Morrow, Peter Scholze

Abstract

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of Breuil-Kisin modules. Notably, this cohomology theory specializes to all other known p-adic cohomology theories, such as crystalline, de Rham and etale cohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings's almost purity theorem, along with a certain functor $Lη$ on the derived category, defined previously by Berthelot-Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham-Witt complexes of Langer-Zink, and can be computed as a q-deformation of de Rham cohomology.

Integral p-adic Hodge theory

Abstract

We construct a new cohomology theory for proper smooth (formal) schemes over the ring of integers of C_p. It takes values in a mixed-characteristic analogue of Dieudonne modules, which was previously defined by Fargues as a version of Breuil-Kisin modules. Notably, this cohomology theory specializes to all other known p-adic cohomology theories, such as crystalline, de Rham and etale cohomology, which allows us to prove strong integral comparison theorems. The construction of the cohomology theory relies on Faltings's almost purity theorem, along with a certain functor on the derived category, defined previously by Berthelot-Ogus. On affine pieces, our cohomology theory admits a relation to the theory of de Rham-Witt complexes of Langer-Zink, and can be computed as a q-deformation of de Rham cohomology.

Paper Structure

This paper contains 57 sections, 157 theorems, 618 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Strategy of the construction
  5. The genesis of this paper
  6. Outline
  7. Acknowledgements
  8. Some examples
  9. A smooth projective surface over $\mathbb Z_2$
  10. An example of degenerating torsion in cohomology
  11. The construction
  12. Algebraic preliminaries on perfectoid rings
  13. Fontaine's ring $A_{\mathrm{inf}}$
  14. Perfectoid rings
  15. The case of a perfectoid field
  16. ...and 42 more sections

Key Result

Theorem 1.1

Let $\mathfrak X$ be a proper smooth formal scheme over $\mathcal{O}_K$, where $\mathcal{O}_K$ is the ring of integers in a complete discretely valued nonarchimedean extension $K$ of $\mathbb Q_p$ with perfect residue field $k$. Let $C$ be a completed algebraic closure of $K$, and write $\mathfrak{X

Theorems & Definitions (236)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6: Fargues
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 226 more