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

TL;DR

This paper develops a method to compute refined localising invariants like $ ext{TC}^-$, revealing new geometric insights and detailed information about homotopy groups that surpasses traditional invariants.

Contribution

It introduces a general recipe for computing refinements of localising invariants and applies it to specific cases involving $ ext{TC}^-$, uncovering new geometric and algebraic information.

Findings

Computed homotopy groups of refined $ ext{TC}^-$ for specific spectra.

Revealed a surprising geometric description of the refined invariants.

Demonstrated that the refined invariants contain non-trivial information modulo any prime.

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 $\operatorname{TC}^-$.