Refined THH Over ku and Habiro Cohomology

By an observation of Efimov and Scholze, Efimov's theorem on the rigidity of localizing motives allows one to construct refinements of topological Hochschild homology and its variants. In this talk, I'll explain how this refinement works, sketch an idea how refined THH over ku should lead to a new cohomology theory for varieties over the rational numbers, and show a few calculations. This new cohomology theory should be able to see a certain part of Scholze's Habiro stack and it should provide the desired modification at the Hodge-Tate point that Scholze's construction is currently missing.