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Prismatic crystals and $q$-Higgs fields

Takeshi Tsuji

Abstract

Similarly to the theory of crystalline cohomology, we give a local description of a prismatic crystal and its cohomology in terms of a $q$-Higgs module and the associated $q$-Higgs complex on the bounded prismatic envelope of an embedding into a framed smooth algebra, when the base bounded prism is defined over the $q$-crystalline prism. We also discuss its behavior under the scalar extension by the Frobenius lifting and tensor products, and a global description of the cohomology of a prismatic crystal via Zariski cohomological descent.

Prismatic crystals and $q$-Higgs fields

Abstract

Similarly to the theory of crystalline cohomology, we give a local description of a prismatic crystal and its cohomology in terms of a -Higgs module and the associated -Higgs complex on the bounded prismatic envelope of an embedding into a framed smooth algebra, when the base bounded prism is defined over the -crystalline prism. We also discuss its behavior under the scalar extension by the Frobenius lifting and tensor products, and a global description of the cohomology of a prismatic crystal via Zariski cohomological descent.
Paper Structure (15 sections, 107 theorems, 233 equations)

This paper contains 15 sections, 107 theorems, 233 equations.

Table of Contents

  1. $\delta$-rings
  2. Divided $\delta$-envelopes
  3. Divided powers on $\delta$-rings
  4. Bounded prisms and bounded prismatic envelopes
  5. Twisted derivations on a ring
  6. Twisted derivations on a $\delta$-ring
  7. Twisted derivations on divided $\delta$-envelopes and Poincaré lemma
  8. Connections
  9. Scalar extensions of integrable connections
  10. $q$-prismatic envelopes and $q$-Higgs fields
  11. Prismatic crystals, stratifications, and $q$-Higgs fields
  12. Linearization and Čech-Alexander complex
  13. Prismatic cohomology and $q$-Higgs complex: Local case
  14. Prismatic cohomology and $q$-Higgs complex: Functoriality
  15. Prismatic cohomology and $q$-Higgs complex: Global case

Key Result

Theorem 1

Let $J$ be an ideal of $R$ containing a power of $pR+[p]_q R$. Then there exists a canonical equivalence between the category of crystals annihilated by $J$ on the prismatic site $(\mathfrak{X}/(R,[p]_q R))_{\hbox{$\mathbbm \Delta$}}$ and that of $D$-modules annihilated by $J$ with quasi-nilpotent $

Theorems & Definitions (265)

  • Theorem 1: Proposition \ref{['prop:CrysStratEquiv']}, Theorem \ref{['thm:StratHiggsEquiv']}
  • Theorem 2: Theorem \ref{['th:CrystalCohqHiggs']}
  • Remark 3
  • Definition 1.9
  • Remark 1.10
  • Proposition 1.11: cf. BS
  • proof
  • Proposition 1.12
  • proof
  • Proposition 2.1: cf. BS
  • ...and 255 more