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$q$-de Rham cohomology and topological Hochschild homology over ku

Ferdinand Wagner

TL;DR

This work develops a $q$-deformation of the Hodge filtration in derived de Rham theory by relating $q$-de Rham cohomology to the even filtration on $TC^-(ku\otimes S_R/ku)$, under the hypothesis that $R$ admits a spherical $\mathbb{E}_2$-lift. It constructs a solid, descent-friendly framework (solid even filtration) to compute these filtrations via even resolutions and trace-class/nuclear theory, and proves $p$-complete and global (adelic) comparisons that identify the $0$-th graded piece with $q$-de Rham cohomology. It then describes Habiro descent through genuine cyclotomic/homotopical structures, expressing the Habiro–Hodge complex as a cyclonic limit of $TC^{-(m)}$-type invariants of $KU_R$ over $KU_A$, and yields a homotopical construction of Habiro rings for number fields. The results unify $q$-de Rham theory with cyclotomic/topological Hochschild homology in a coherent global framework and provide explicit filtrations and base-change formulas essential for arithmetic applications.

Abstract

Hodge-filtered derived de Rham cohomology of a ring $R$ can be described (up to completion and shift) as the graded pieces of the even filtration on $\mathrm{HC}^-(R)$. In this paper we show a deformation of this result: If $R$ admits a spherical $\mathbb{E}_2$-lift, then the graded pieces of the even filtration on $\mathrm{TC}^-(\mathrm{ku}\otimes\mathbb{S}_R/\mathrm{ku})$ form a certain filtration on the $q$-de Rham cohomology of $R$, which $q$-deforms the Hodge filtration. We also explain how the associated Habiro-Hodge complex can be described in terms of the genuine equivariant structure on $\mathrm{THH}(\mathrm{KU}\otimes\mathbb{S}_R/\mathrm{KU})$. As a special case, we'll obtain homotopy-theoretic construction of the Habiro ring of a number field of Garoufalidis-Scholze-Wheeler-Zagier.

$q$-de Rham cohomology and topological Hochschild homology over ku

TL;DR

This work develops a -deformation of the Hodge filtration in derived de Rham theory by relating -de Rham cohomology to the even filtration on , under the hypothesis that admits a spherical -lift. It constructs a solid, descent-friendly framework (solid even filtration) to compute these filtrations via even resolutions and trace-class/nuclear theory, and proves -complete and global (adelic) comparisons that identify the -th graded piece with -de Rham cohomology. It then describes Habiro descent through genuine cyclotomic/homotopical structures, expressing the Habiro–Hodge complex as a cyclonic limit of -type invariants of over , and yields a homotopical construction of Habiro rings for number fields. The results unify -de Rham theory with cyclotomic/topological Hochschild homology in a coherent global framework and provide explicit filtrations and base-change formulas essential for arithmetic applications.

Abstract

Hodge-filtered derived de Rham cohomology of a ring can be described (up to completion and shift) as the graded pieces of the even filtration on . In this paper we show a deformation of this result: If admits a spherical -lift, then the graded pieces of the even filtration on form a certain filtration on the -de Rham cohomology of , which -deforms the Hodge filtration. We also explain how the associated Habiro-Hodge complex can be described in terms of the genuine equivariant structure on . As a special case, we'll obtain homotopy-theoretic construction of the Habiro ring of a number field of Garoufalidis-Scholze-Wheeler-Zagier.

Paper Structure

This paper contains 31 sections, 83 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. q-Hodge filtrations from THH over ku
  3. Habiro descent, homotopically
  4. Organisation of this paper
  5. The solid even filtration
  6. Definitions and basic properties
  7. Recollections on trace-class morphisms and nuclear objects
  8. Solid even flatness in the nuclear case
  9. Solid faithfully flat descent in the nuclear case
  10. The solid even filtration for THH
  11. Solid THH
  12. The solid even filtration via even resolutions
  13. Base change
  14. Comparison of even filtrations
  15. q-de Rham cohomology and TC- over ku
  16. ...and 16 more sections

Key Result

Theorem 1

Let $R$ be a quasi-syntomic ring. Then completion of the Hodge-filtrered derived de Rham complex of $R$ agrees (up to shift) with the the graded pieces of the $S^1$-equivariant even filtration on $\mathop{\mathrm{\mathrm{HC}}}\nolimits^-(R/\mathbb{Z})$:

Theorems & Definitions (191)

  • Theorem 1: Antieau--Hahn--Raksit--Wilson
  • Theorem 2: see \ref{['thm:qdeRhamkuGlobal']}
  • Theorem 4: Raksit, unpublished; see \ref{['thm:Raksit']}
  • Theorem 6: Devalapurkar DevalapurkarSpherochromatism
  • Theorem 14: see \ref{['thm:qdeRhamKUGenuine']}
  • Corollary 15: see \ref{['cor:HabiroRingOfNumberFieldKU']}
  • Lemma 26
  • proof
  • Theorem 28
  • proof : Proof sketch
  • ...and 181 more