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A motivic filtration on the topological cyclic homology of commutative ring spectra

Jeremy Hahn, Arpon Raksit, Dylan Wilson

TL;DR

This work develops a global, functorial even filtration on THH and related cyclotomic invariants, extending Bhatt–Morrow–Scholze's motivic filtration to a broad class of commutative ring spectra via an alternate construction that interacts with cyclotomic structure and descent. For chromatically quasisyntomic E_∞-rings, the authors define motivic filtrations on THH, TC^-, TP, and TC, showing convergence and compatibility with Frobenius maps; these filtrations recover prismatic and syntomic cohomology in the associated graded, and they provide a powerful framework for computation. A central computational achievement is the mod $(p,v_1)$ syntomic cohomology of the Adams summand ell, described as a finite $F_p[v_2]$-module with explicit Adams-weight generators, and the collapse of the corresponding motivic spectral sequence for V(1)_*TC(ell) when $p eq 2$ (and refined structure at $p=3$). The results connect to HKR, Morin–Bhatt–Lurie motivic filtrations, and prismatic cohomology, offering a robust, global toolkit for understanding THH/TC in chromatic settings and enabling practical calculations of algebraic K-theory via motivic spectral sequences.

Abstract

For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page.

A motivic filtration on the topological cyclic homology of commutative ring spectra

TL;DR

This work develops a global, functorial even filtration on THH and related cyclotomic invariants, extending Bhatt–Morrow–Scholze's motivic filtration to a broad class of commutative ring spectra via an alternate construction that interacts with cyclotomic structure and descent. For chromatically quasisyntomic E_∞-rings, the authors define motivic filtrations on THH, TC^-, TP, and TC, showing convergence and compatibility with Frobenius maps; these filtrations recover prismatic and syntomic cohomology in the associated graded, and they provide a powerful framework for computation. A central computational achievement is the mod syntomic cohomology of the Adams summand ell, described as a finite -module with explicit Adams-weight generators, and the collapse of the corresponding motivic spectral sequence for V(1)_*TC(ell) when (and refined structure at ). The results connect to HKR, Morin–Bhatt–Lurie motivic filtrations, and prismatic cohomology, offering a robust, global toolkit for understanding THH/TC in chromatic settings and enabling practical calculations of algebraic K-theory via motivic spectral sequences.

Abstract

For a prime number and a -quasisyntomic commutative ring , Bhatt--Morrow--Scholze defined motivic filtrations on the -completions of and , with the associated graded objects for and recovering the prismatic and syntomic cohomology of , respectively. We give an alternate construction of these filtrations that applies also when is a well-behaved commutative ring spectrum; for example, we can take to be , , , , or . We compute the mod syntomic cohomology of the Adams summand and observe that, when , the motivic spectral sequence for collapses at the -page.
Paper Structure (29 sections, 82 theorems, 168 equations)

This paper contains 29 sections, 82 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. The even filtration and comparison theorems
  3. Variants of the even filtration
  4. $\mathrm{THH}$ of chromatically quasisyntomic rings
  5. The motivic spectral sequence for $V(1)_\star \mathrm{TC}(\ell)$
  6. Acknowledgments
  7. Conventions
  8. The even filtration
  9. Defining the filtrations
  10. Descent properties of the filtrations
  11. The underlying objects of the filtrations
  12. $p$-Completion and rationalization
  13. Wilson spaces
  14. Preliminaries
  15. A sufficient supply of cyclotomic bases
  16. ...and 14 more sections

Key Result

Theorem 1.1.3

Let $R$ be a discrete commutative ring with bounded $p$-power torsion for all primes $p$ and such that the algebraic cotangent complex $\mathrm{L}^\mathrm{alg}_R$ has Tor-amplitude contained in $[0,1]$. Then there is a canonical equivalence where $\operatorname{fil}^\star_{\mathrm{mot}}$ denotes the global motivic filtration of Morin and Bhatt--Lurie.

Theorems & Definitions (230)

  • Definition 1.1.1
  • Remark 1.1.2
  • Theorem 1.1.3
  • Remark 1.1.4
  • Theorem 1.1.5
  • Corollary 1.1.6
  • Remark 1.1.7
  • Theorem 1.1.8
  • Remark 1.1.9
  • Definition 1.2.1
  • ...and 220 more