q-Hodge filtrations, Habiro cohomology, and ku

Abstract

Peter Scholze has raised the question whether some variant of the q-de Rham complex is already defined over the Habiro ring. Such a variant should then be called Habiro cohomology. <br /> In Part I of this thesis we show that Habiro cohomology exists whenever the q-de Rham complex can be equipped with a q-Hodge filtration: a q-deformation of the Hodge filtration, subject to some reasonable conditions. To any such q-Hodge filtration we associate a small modification of the q-de Rham complex, which we call the q-Hodge complex, and show that it descends canonically to the Habiro ring. This construction recovers and generalises the Habiro ring of a number field from work of Garoufalidis-Scholze-Wheeler-Zagier and it is closely related to the q-de Rham-Witt complexes from previous work of the author. <br /> While there is no canonical q-Hodge filtration in general, we show that such a filtration does exist in many cases of interest. For example, for a smooth scheme X over the integers, a canonical choice of q-Hodge filtration does exist as soon as one inverts all primes up to the dimension of X. <br /> In Part II, we explain how another large class of examples arises from homotopy theory: If R is a quasisyntomic ring which admits an E₂-lift to the sphere spectrum, one can use the even filtration on topological negative cyclic homology over the complex K-theory spectra ku and KU to obtain a q-Hodge filtration and the associated q-Hodge complex for R. We also explain how the Habiro descent of the q-Hodge complex can be recovered using genuine equivariant homotopy theory. <br /> In Part III, which is based on joint work with Samuel Meyer, we study a refinement of topological Hochschild/negative cyclic homology (THH/TC^-), constructed by Efimov and Scholze as a consequence of Efimov's theorem on the rigidity of localising motives. Using the results from Part II, we'll compute refined TC^- in an interesting special case.