$q$-Hodge complexes and refined $\operatorname{TC}^-$
Samuel Meyer, Ferdinand Wagner
TL;DR
The paper develops a general framework to refine localising invariants such as THH and TC^- using Efimov's rigidity, giving a computable recipe via towers of E1-algebras and killing pro-idempotents. Applying this to ku and KU with rational inputs, it proves that TC^{-,ref}((ku⊗Q)/ku) and TC^{-,ref}((KU⊗Q)/KU) are concentrated in even degrees and identifies their coefficient algebras with explicit q-Hodge-theoretic objects: A_{ku}^* and A_{KU}, realized geometrically as overconvergent function algebras on analytic spaces Z^{*,dagger}/T and Z^dagger. The constructions yield concrete descriptions in terms of q-Hodge filtrations on derived q-de Rham complexes and connect to Habiro-type descent, with potential for extending to higher chromatic bases. The work also provides explicit generators for the q-Hodge filtrations, enabling the overconvergent descriptions and paving the way for q-de Rham–style cohomology theories RΓ_{ku} and RΓ_{KU} for rational varieties.
Abstract
As a consequence of Efimov's proof of rigidity of the $\infty$-category of localising motives, Efimov and Scholze have constructed refinements of localising invariants such as $\operatorname{THH}$ and $\operatorname{TC}^-$. These refinements often contain vastly more information than the original invariant. In this article we explain a general recipe how to compute the refinements in certain situations. We then apply this recipe to compute the homotopy groups of $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{ku}\otimes\mathbb Q/\mathrm{ku})$ and $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{KU}\otimes\mathbb Q/\mathrm{KU})$. The result has a rather surprising geometric description and contains non-trivial information modulo any prime, in contrast to the unrefined $\operatorname{TC}^-$.