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

As a consequence of Efimov’s proof of rigidity of the $\mathrm{\infty }$-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 ${\mathrm{TC}}^{-,\mathrm{r}\mathrm{e}\mathrm{f}}(\mathrm{k}\mathrm{u}\otimes \mathbb{Q}\mathrm{/}\mathrm{k}\mathrm{u})$ and ${\mathrm{TC}}^{-,\mathrm{r}\mathrm{e}\mathrm{f}}(\mathrm{K}\mathrm{U}\otimes \mathbb{Q}\mathrm{/}\mathrm{K}\mathrm{U})$. The result has a rather surprising geometric description and contains non-trivial information modulo any prime, in contrast to the unrefined ${\mathrm{TC}}^{-}$.