Thomason filtration via $T(1)$-local $\mathrm{TC}$

TL;DR

This work constructs a prismatic, descent-based filtration on $T(1)$-local TC for animated commutative rings, providing explicit associated-graded pieces governed by étale cohomology and a refined arc$_{p}$-hypersheaf version. The filtration is exhaustive, multiplicative, and compatible with sifted colimits and $p$-complete descent, and it recovers Thomason’s spectral sequence for $T(1)$-local algebraic K-theory via the cyclotomic trace. The authors also develop pro-Galois descent at the generic fiber and establish étale comparisons for prismatic cohomology, thereby connecting K-theory, TC, and prismatic cohomology in a unified, filtrated framework. Their approach yields a K-theoretic route to étale comparison and clarifies how Nygaard-type filtrations interact with Frobenius fixed points in the $T(1)$-local context, with descent control provided by arc$_{p}$-descent. Overall, the results offer a robust, filtration-enhanced paradigm for studying $p$-adic cohomology theories via noncommutative motives and derived descent.

Abstract

We construct a natural filtration on $T(1)$-local $\mathrm{TC}$ for any animated commutative rings using prismatic cohomology and descent theory. In the course of the construction, we also study some general properties of prismatic cohomology complexes over perfect prisms after inverting distinguished generators. The construction is intrinsic to $\mathrm{TC}$ and recovers Thomason's spectral sequence for $T(1)$-local algebraic K-theory via the cyclotomic trace map; as a consequence, we also recover the étale comparison for prismatic cohomology.

Theorems & Definitions (92)

• Theorem 1.1: Theorem \ref{['th:statementT(1)TCfilt']}
• Remark 1.2
• Remark 1.3
• Theorem 1.4: Theorem \ref{['th:statementfiltonarcpsheaf']}
• Theorem 1.5: Theorem \ref{['th:statementcompwiththomason']}
• Corollary 1.6: Corollary \ref{['cor:etcomp']}
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof