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Revisiting derived crystalline cohomology

Zhouhang Mao

Abstract

We prove that the $\infty$-category of surjections of animated rings is projectively generated, introduce and study the notion of animated PD-pairs - surjections of animated rings with a "derived" PD-structure. This allows us to generalize classical results to non-flat and non-finitely-generated situations. Using animated PD-pairs, we develop several approaches to derived crystalline cohomology and establish comparison theorems. As an application, we generalize the comparison between derived and classical crystalline cohomology from syntomic (affine) schemes (due to Bhatt) to quasisyntomic schemes. We also develop a non-completed animated analogue of prisms and prismatic envelopes. We prove a variant of the Hodge-Tate comparison for animated prismatic envelopes from which we deduce a result about flat cover of the final object for quasisyntomic schemes, which generalizes several known results under smoothness and finiteness conditions.

Revisiting derived crystalline cohomology

Abstract

We prove that the -category of surjections of animated rings is projectively generated, introduce and study the notion of animated PD-pairs - surjections of animated rings with a "derived" PD-structure. This allows us to generalize classical results to non-flat and non-finitely-generated situations. Using animated PD-pairs, we develop several approaches to derived crystalline cohomology and establish comparison theorems. As an application, we generalize the comparison between derived and classical crystalline cohomology from syntomic (affine) schemes (due to Bhatt) to quasisyntomic schemes. We also develop a non-completed animated analogue of prisms and prismatic envelopes. We prove a variant of the Hodge-Tate comparison for animated prismatic envelopes from which we deduce a result about flat cover of the final object for quasisyntomic schemes, which generalizes several known results under smoothness and finiteness conditions.

Paper Structure

This paper contains 30 sections, 198 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Background and main results
  3. Main techniques
  4. Main definitions and constructions
  5. Structure of article
  6. Notations and terminology
  7. Categorical preparations
  8. Animation of adjoint functors
  9. Diagram categories and undercategories
  10. Comma categories and base change
  11. $\infty$-category of graded and filtered objects
  12. Reflective subcategories
  13. Animated ideals and PD-pairs
  14. Smith ideals
  15. Animated (PD-)pairs
  16. ...and 15 more sections

Key Result

Theorem 1.1

The $\infty$-category of surjective maps of animated rings is projectively generated. The set $\{ \mathbb{Z} [x_1, \ldots, x_m, y_1, \ldots, y_n] \twoheadrightarrow \mathbb{Z} [x_1, \ldots, x_m] | m, n \in \mathbb{N} \}$ of objects forms a set of compact projective generators.

Theorems & Definitions (403)

  • Theorem 1.1: \ref{['thm:ani-smith-eq']}
  • Theorem 1.2: \ref{['cor:conjfil-gr', 'cor:LAdFil-symm-cot-cx']}
  • Theorem 1.3: \ref{['cor:Fp-qreg-frob-ani-pd-env']}
  • Theorem 1.4: \ref{['prop:koszul-regular-ani-pd-env']}
  • Theorem 1.5: \ref{['prop:comp-PDFil-equiv-qreg', 'prop:qreg-assoc-gr-pd']}
  • Theorem 1.6: \ref{['prop:crys-PD-env-equiv']}
  • Theorem 1.7: \ref{['prop:crys-Cartier-isom']}
  • Theorem 1.8: \ref{['prop:deriv-crys-coh-site-comp', 'prop:site-comp-qsyn-integral', 'prop:site-static-comp']}
  • Theorem 1.9: \ref{['thm:Hdg-Tate']}
  • Theorem 1.10: \ref{['prop:compl-prism-flat-cover']}
  • ...and 393 more