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$q$-Hodge complexes over the Habiro ring

Ferdinand Wagner

TL;DR

This work constructs a canonical descent mechanism for $q$-Hodge data to the Habiro ring, showing that a $q$-Hodge filtration on the derived $q$-de Rham complex yields a $q$-Hodge complex that descends to the Habiro ring $\\mathcal{H}$. It develops a robust framework linking Habiro descent, $q$-de Rham–Witt theory, and Nygaard-type filtrations via twisted $q$-de Rham complexes, establishing a Habiro-Hodge complex that canonically recovers truncated $q$-de Rham–Witt objects and interacts monoidally with derived commutative structures. The paper also provides constructive, functorial filtrations in large subcategories (e.g., smooth algebras with primes inverted up to a bound) and extends the theory to quasi-regular quotients, highlighting both the algebraic and analytic sides of Habiro cohomology. Overall, it clarifies when and how Habiro cohomology can be defined and computed, and it situates Habiro descent as a natural enhancement of $q$-de Rham/Witt theory with deep connections to regulator phenomena and potential Scholze analytic Habiro geometry.

Abstract

Peter Scholze has raised the question whether some variant of the $q$-de Rham complex is already defined over the Habiro ring $\mathcal H = \lim_{m\in\mathbb N}\mathbb Z[q]_{(q^m-1)}^\wedge$. We show that such a variant exists whenever the $q$-de Rham complex can be equipped with a "$q$-Hodge filtration": a $q$-deformation of the Hodge filtration, subject to some reasonable conditions. To any such $q$-Hodge filtration we associate a small modification of the $q$-de Rham complex, which we call the $q$-Hodge complex, and show that it descends canonically to the Habiro ring. This construction recovers and generalises the Habiro ring of a number field of Garoufalidis-Scholze-Wheeler-Zagier and is closely related to the $q$-de Rham--Witt complexes from previous work of the author as well as, conjecturally, to Scholze's analytic Habiro stack. While there's no canonical $q$-Hodge filtration in general, we show that it does exist in many cases of interest. For example, for a smooth scheme $X$ over $\mathbb Z$, the $q$-de Rham complex can be equipped with a canonical $q$-Hodge filtration as soon as one inverts all primes $p\leq \dim(X/\mathbb Z)$.

$q$-Hodge complexes over the Habiro ring

TL;DR

This work constructs a canonical descent mechanism for -Hodge data to the Habiro ring, showing that a -Hodge filtration on the derived -de Rham complex yields a -Hodge complex that descends to the Habiro ring . It develops a robust framework linking Habiro descent, -de Rham–Witt theory, and Nygaard-type filtrations via twisted -de Rham complexes, establishing a Habiro-Hodge complex that canonically recovers truncated -de Rham–Witt objects and interacts monoidally with derived commutative structures. The paper also provides constructive, functorial filtrations in large subcategories (e.g., smooth algebras with primes inverted up to a bound) and extends the theory to quasi-regular quotients, highlighting both the algebraic and analytic sides of Habiro cohomology. Overall, it clarifies when and how Habiro cohomology can be defined and computed, and it situates Habiro descent as a natural enhancement of -de Rham/Witt theory with deep connections to regulator phenomena and potential Scholze analytic Habiro geometry.

Abstract

Peter Scholze has raised the question whether some variant of the -de Rham complex is already defined over the Habiro ring . We show that such a variant exists whenever the -de Rham complex can be equipped with a "-Hodge filtration": a -deformation of the Hodge filtration, subject to some reasonable conditions. To any such -Hodge filtration we associate a small modification of the -de Rham complex, which we call the -Hodge complex, and show that it descends canonically to the Habiro ring. This construction recovers and generalises the Habiro ring of a number field of Garoufalidis-Scholze-Wheeler-Zagier and is closely related to the -de Rham--Witt complexes from previous work of the author as well as, conjecturally, to Scholze's analytic Habiro stack. While there's no canonical -Hodge filtration in general, we show that it does exist in many cases of interest. For example, for a smooth scheme over , the -de Rham complex can be equipped with a canonical -Hodge filtration as soon as one inverts all primes .

Paper Structure

This paper contains 24 sections, 61 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Habiro cohomology
  3. q-Hodge filtrations and Habiro descent
  4. Existence results for q-Hodge filtrations
  5. Organisation of this paper
  6. Habiro rings of étale extensions
  7. A general descent principle
  8. Habiro rings of étale extensions
  9. Habiro descent for q-Hodge complexes
  10. q-Hodge filtrations and the q-Hodge complex
  11. The main result
  12. Deformations of q-de Rham--Witt complexes
  13. The Nygaard filtration on q-de Rham--Witt complexes
  14. The twisted q-Hodge filtration
  15. Habiro descent for q-Hodge complexes
  16. ...and 9 more sections

Key Result

Theorem 11

Let $\widehat{\mathcal{D}}_{(q-1)}(\mathbb{Z}[q])$ and $\widehat{\mathcal{D}}_\mathcal{H}(\mathbb{Z}[q])$ denote the $(q-1)$- and Habiro-complete objects (see appendix:HabiroCompletion), respectively, in the derived $\infty$-category of $\mathbb{Z}[q]$.

Theorems & Definitions (152)

  • Definition 6
  • Remark 7
  • Remark 8
  • Theorem 11: see \ref{['thm:HabiroDescent']}
  • Definition 12
  • Remark 13
  • Theorem 15: see \ref{['thm:CanonicalqHodgeSmooth']}
  • Theorem 19: see \ref{['thm:qHodgeWellBehaved']}
  • Theorem 20: see \ref{['thm:CanonicalqHodgeQuasiregular']}
  • Lemma 25
  • ...and 142 more