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On endomorphisms of the de Rham cohomology functor

Shizhang Li, Shubhodip Mondal

Abstract

We compute the moduli of endomorphisms of the de Rham and crystalline cohomology functors, viewed as a cohomology theory on smooth schemes over truncated Witt vectors. As applications of our result, we deduce Drinfeld's refinement of the classical Deligne--Illusie decomposition result for de Rham cohomology of varieties in characteristic $p>0$ that are liftable to $W_2$, and prove further functorial improvements.

On endomorphisms of the de Rham cohomology functor

Abstract

We compute the moduli of endomorphisms of the de Rham and crystalline cohomology functors, viewed as a cohomology theory on smooth schemes over truncated Witt vectors. As applications of our result, we deduce Drinfeld's refinement of the classical Deligne--Illusie decomposition result for de Rham cohomology of varieties in characteristic that are liftable to , and prove further functorial improvements.

Paper Structure

This paper contains 17 sections, 45 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Stacky approach to de Rham cohomology
  3. Quasi-ideals
  4. Ring stacks
  5. Affine stacks
  6. Unwinding ring stacks
  7. de Rham cohomology via unwinding
  8. Endomorphisms of de Rham cohomology I
  9. Endomorphisms of de Rham cohomology II
  10. Unwinding equivalence
  11. Construction of endomorphisms
  12. Calculation of the endomorphism monoid
  13. Application to the Deligne--Illusie decomposition
  14. Drinfeld's refinement of the Deligne--Illusie decomposition
  15. Uniqueness of functorial splittings
  16. ...and 2 more sections

Key Result

Theorem 1.1

Theorems & Definitions (135)

  • Theorem 1.1: Special case of Main \ref{['representing endomorphism monoid']}
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Special case of \ref{['integral endomorphism prop']}
  • Remark 1.5
  • Theorem 1.6: Drinfeld, BhaLur2, see \ref{['Drinfeld splitting']}
  • Corollary 1.7: see \ref{['no further functorial splitting']}
  • Theorem 1.8: see \ref{['uniqueness of functorial splitting of Conj1']} for the precise statement
  • Remark 1.9
  • Definition 2.1: Quasi-ideals
  • ...and 125 more