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Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator

Alexander Petrov

Abstract

We describe the obstruction to decomposing in degrees $\leq p$ the de Rham complex of a smooth variety over a perfect field $k$ of characteristic $p$ that lifts over $W_2(k)$, and show that there exist liftable smooth projective varieties of dimension $p+1$ whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. We also describe the action of the Sen operator on the de Rham complex in degrees $\leq p$ and give examples of varieties with a non-semisimple Sen operator. Our methods rely on the commutative algebra structure on de Rham and Hodge-Tate cohomology, and are inspired by the properties of Steenrod operations on cohomology of cosimplicial commutative algebras. The example of a non-degenerate Hodge-to-de Rham spectral sequence relies on a non-vanishing result on cohomology of groups of Lie type. We give applications to other situations such as describing extensions in the canonical filtration on de Rham, Hodge, and étale cohomology of an abelian variety equipped with a group action. We also show that the de Rham complex of a smooth variety over $k$ is formal as an $E_{\infty}$-algebra if and only if the variety lifts to $W_2(k)$ together with its Frobenius endomorphism.

Non-decomposability of the de Rham complex and non-semisimplicity of the Sen operator

Abstract

We describe the obstruction to decomposing in degrees the de Rham complex of a smooth variety over a perfect field of characteristic that lifts over , and show that there exist liftable smooth projective varieties of dimension whose Hodge-to-de Rham spectral sequence does not degenerate at the first page. We also describe the action of the Sen operator on the de Rham complex in degrees and give examples of varieties with a non-semisimple Sen operator. Our methods rely on the commutative algebra structure on de Rham and Hodge-Tate cohomology, and are inspired by the properties of Steenrod operations on cohomology of cosimplicial commutative algebras. The example of a non-degenerate Hodge-to-de Rham spectral sequence relies on a non-vanishing result on cohomology of groups of Lie type. We give applications to other situations such as describing extensions in the canonical filtration on de Rham, Hodge, and étale cohomology of an abelian variety equipped with a group action. We also show that the de Rham complex of a smooth variety over is formal as an -algebra if and only if the variety lifts to together with its Frobenius endomorphism.
Paper Structure (39 sections, 110 theorems, 170 equations)

This paper contains 39 sections, 110 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. The main results.
  3. Proof of Theorem \ref{['intro: main extension class']}.
  4. Liftable variety with a non-degenerate conjugate spectral sequence
  5. Computation of the Sen operator.
  6. Preliminaries on homotopical algebra
  7. Non-abelian derived functors.
  8. Derived commutative algebras.
  9. Cosimplicial commutative algebras.
  10. Bockstein morphisms.
  11. Symmetric power , class , and Steenrod operations
  12. Symmetric powers vs. divided powers.
  13. $T_p$ and commutative algebras.
  14. Steenrod operations on cohomology of cosimplicial algebras.
  15. Steenrod operations on cohomology of $E_{\infty}$-algebras.
  16. ...and 24 more sections

Key Result

Theorem 1.1

There exists a smooth projective variety $X_0$ over $k$ of dimension $p+1$ that lifts to $W(k)$ such that $\dim_k H^p_{\mathrm{dR}}(X_0/k)<\sum\limits_{i+j=p}\dim_k H^j(X_0,\Omega^i_{X_0/k})$. In particular, the Hodge-to-de Rham spectral sequence of $X_0$ does not degenerate at the first page, and $

Theorems & Definitions (221)

  • Theorem 1.1: Corollary \ref{['nondeg example: main corollary']}
  • Theorem 1.2: Theorem \ref{['cosimp applications: the best part de Rham']} if $X_1$ lifts to $W(k)$, Corollary \ref{['semiperf: smooth cor']} in general
  • Theorem 1.3: Theorem \ref{['semiperf: main sen operator']}
  • Corollary 1.4: Proposition \ref{['nonsemisimp: p dim example']}
  • Theorem 1.5: Theorem \ref{['cosimp: main theorem']}
  • Proposition 1.6: Proposition \ref{['free cosimplicial: steenrod operations description prop']}
  • Proposition 1.7: Proposition \ref{['cosimp applications: de rham formality']}
  • Proposition 1.8: Proposition \ref{['rational group cohomology: main non-vanishing']}
  • Proposition 1.9: Proposition \ref{['nondeg example: AV quotient']}
  • Proposition 1.10: Corollary \ref{['AV coherent: supersingularity criterion']}
  • ...and 211 more