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$q$-Witt vectors and $q$-Hodge complexes

Ferdinand Wagner

TL;DR

This work develops a rigorous $q$-variant of Witt vectors and de Rham–Witt complexes, tying it to Habiro rings, $q$-Hodge theory, and THH(-/ku). It defines truncated big $q$-Witt vectors, their relative versions, ghost maps, and Teichmüller lifts, and then builds $q$-de Rham–Witt complexes with a Frobenius structure and étale-base-change compatibility. In the smooth case it constructs a $q$-Hodge complex and proves a $p$-typical comparison isomorphism between $\operatorname{q}$-$\mathbb{W}_m\Omega_{R/A}^*$ and the $q$-Hodge cohomology, with global extensions via arithmetic fracture squares. The paper culminates in a no-go result for functoriality of the $q$-Hodge complex in general, motivating partial functorial constructions and future work (qWittHabiro) linking to THH and Habiro-type cohomology.

Abstract

In this article, we'll introduce a $q$-variant of Witt vectors and de Rham-Witt complexes. This variant is closely related to the Habiro ring of a number field constructed by Garoufalidis, Scholze, Wheeler, and Zagier, to $q$-Hodge cohomology, and to $\operatorname{THH}(-/\mathrm{ku})$. While most of these connections will only be explored in forthcoming work, the goal of this article is to provide the necessary technical foundation.

$q$-Witt vectors and $q$-Hodge complexes

TL;DR

This work develops a rigorous -variant of Witt vectors and de Rham–Witt complexes, tying it to Habiro rings, -Hodge theory, and THH(-/ku). It defines truncated big -Witt vectors, their relative versions, ghost maps, and Teichmüller lifts, and then builds -de Rham–Witt complexes with a Frobenius structure and étale-base-change compatibility. In the smooth case it constructs a -Hodge complex and proves a -typical comparison isomorphism between - and the -Hodge cohomology, with global extensions via arithmetic fracture squares. The paper culminates in a no-go result for functoriality of the -Hodge complex in general, motivating partial functorial constructions and future work (qWittHabiro) linking to THH and Habiro-type cohomology.

Abstract

In this article, we'll introduce a -variant of Witt vectors and de Rham-Witt complexes. This variant is closely related to the Habiro ring of a number field constructed by Garoufalidis, Scholze, Wheeler, and Zagier, to -Hodge cohomology, and to . While most of these connections will only be explored in forthcoming work, the goal of this article is to provide the necessary technical foundation.

Paper Structure

This paper contains 21 sections, 74 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. q-Witt vectors
  3. Some technical preliminaries
  4. Definitions and basic properties
  5. Injectivity of W(R) -> q-W(R)
  6. q-Witt vectors of Lambda-rings
  7. Relative q-Witt vectors
  8. q-Witt vectors and étale morphisms
  9. q-de Rham--Witt complexes
  10. Three non-equivalent categories
  11. Construction of q-de Rham--Witt complexes
  12. Construction of Frobenii
  13. Étale base change
  14. q-de Rham Witt complexes in the smooth case
  15. p-Torsion freeness of q-de Rham--Witt complexes
  16. ...and 6 more sections

Key Result

Theorem 5

If $A$ is equipped with a $\Lambda$-structure and $\mathbb{Z}$-torsion free, then there exists a functor from the category of smooth $A$-algebras into the $\infty$-category of $(q-1)$-complete $\mathbb{E}_\infty$-algebras over $A\llbracket q-1\rrbracket$, such that $\operatorname{\mathnormal{q}}\!\space\mhyph\space\Omega_{-/A}/(q-1)\simeq \Omega_{-/A}$ agrees with the de Rham complex functor and

Theorems & Definitions (180)

  • Remark 2
  • Theorem 5: see Prismatic for the essential case
  • Theorem 7: see \ref{['thm:qDeRhamWittqHodge']}
  • Remark 8
  • Theorem 9: see \ref{['thm:qHodgeNotFunctorial']}
  • Lemma 13
  • proof
  • Lemma 14
  • proof
  • Lemma 15
  • ...and 170 more