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Ogus-Vologodsky equivalence via stacks

Gleb Terentiuk

Abstract

Using the relative de Rham stack for a family $X \to S$ in characteristic $p,$ we reprove the (local and global) Ogus-Vologodsky equivalence. Moreover, we observe that a lift of $S$ is not necessary. Instead, we use a lift of $X$ to the second Witt vectors of $S.$ The main ingredient is that, for a quasi-syntomic family $X/S,$ the relative de Rham stack admits a structure of a torsor over $X'$ which is the analogue of the Azumaya property of the algebra of differential operators. This can be applied to families of (reasonable) algebraic stacks, which gives rise to a logarithmic version of the Cartier equivalence. Along the way, we also obtain a decompleted version of the global Cartier equivalence.

Ogus-Vologodsky equivalence via stacks

Abstract

Using the relative de Rham stack for a family in characteristic we reprove the (local and global) Ogus-Vologodsky equivalence. Moreover, we observe that a lift of is not necessary. Instead, we use a lift of to the second Witt vectors of The main ingredient is that, for a quasi-syntomic family the relative de Rham stack admits a structure of a torsor over which is the analogue of the Azumaya property of the algebra of differential operators. This can be applied to families of (reasonable) algebraic stacks, which gives rise to a logarithmic version of the Cartier equivalence. Along the way, we also obtain a decompleted version of the global Cartier equivalence.

Paper Structure

This paper contains 27 sections, 72 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Local Cartier transform
  3. Powers of Frobenii
  4. Applications.
  5. Conventions.
  6. Acknowledgments
  7. Preliminaries
  8. Animated rings
  9. Ring stacks
  10. Derived rings and affine stacks
  11. Relative de Rham stack
  12. Preliminaries on de Rham stack
  13. Affineness
  14. Splittings of the relative de Rham gerbe
  15. Quasi-coherent sheaves on de Rham stack
  16. ...and 12 more sections

Key Result

Theorem 1.1

(MR2373230) Let $X \to S$ be a smooth morphism. Any lift of the relative Frobenius $X \to X'$ to a flat lift $\tilde{S} \to {\mathbf Z}/p^2$ induces a symmetric monoidal equivalence $\mathop{\mathrm{HIG}}\nolimits^{\cdot}(X'/S) \simeq \mathop{\mathrm{MIC}}\nolimits^{\cdot}(X/S).$$\blacktriangleleft$

Theorems & Definitions (224)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1
  • proof
  • ...and 214 more