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On log crystalline higher direct image

Heng Du, Yong Suk Moon, Koji Shimizu

Abstract

We define the big crystalline site for a log scheme and prove the basic properties. In particular, we show the boundedness, base change, and perfectness theorems for the crystalline higher direct image of quasi-coherent crystals between fine log schemes. We also introduce the big absolute crystalline sites and discuss the Frobenius isogeny property of the crystalline higher direct image of $F$-isocrystals.

On log crystalline higher direct image

Abstract

We define the big crystalline site for a log scheme and prove the basic properties. In particular, we show the boundedness, base change, and perfectness theorems for the crystalline higher direct image of quasi-coherent crystals between fine log schemes. We also introduce the big absolute crystalline sites and discuss the Frobenius isogeny property of the crystalline higher direct image of -isocrystals.
Paper Structure (11 sections, 48 theorems, 112 equations)

This paper contains 11 sections, 48 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Log PD-envelopes
  3. Definition of big log crystalline sites
  4. Functoriality of crystalline topoi
  5. Projections to small étale topoi
  6. Embedding into a smooth log scheme
  7. Boundedness of log crystalline cohomology
  8. Base change
  9. Crystalline derived pushforward of finite locally free modules
  10. Big crystalline sites in the p-adic base and absolute cases
  11. Frobenius isogeny property

Key Result

Theorem 1.1

Let $f\colon (X,M_X)\rightarrow (Y,M_Y)$ be a morphism of fine log $\Sigma^\sharp$-schemes. The same properties hold for small crystalline site.

Theorems & Definitions (121)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Theorem \ref{['thm:Frobenius isogeny property']}, Remark \ref{['rem:Frobenius isogeny property']}
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • ...and 111 more