$q$-Hodge complexes and refined ${\mathrm{TC}}^{-}$

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 $\mathrm{THH}$ and ${\mathrm{TC}}^{-}$. 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 ${\mathrm{TC}}^{-,\mathrm{ref}}(\mathrm{ku}\otimes \mathbb{Q}/\mathrm{ku})$ and ${\mathrm{TC}}^{-,\mathrm{ref}}(\mathrm{KU}\otimes \mathbb{Q}/\mathrm{KU})$. The result has a rather surprising geometric description and contains non-trivial information modulo any prime, in contrast to the unrefined ${\mathrm{TC}}^{-}$.

arXiv comment57 pages, 1 figure. Rewrite of the original version: The material about q-Hodge filtrations has been removed and turned into a separate paper. A section on general computation of refined invariants has been added