Ferdinand Wagner  (Universität Bonn): q-Hodge filtrations, Habiro cohomology, and THH over ku

The recent work of Garoufalidis, Scholze, Wheeler, and Zagier on the “Habiro ring of a number field” has sparked the question whether there exists a cohomology theory for smooth schemes over Z with coefficients in the Habiro ring, the completion of Z[q] at all roots of unity, and with specialisations to étale and de Rham cohomology. In this talk I’ll explain how this question is intimately related to the question whether the Hodge filtration on de Rham cohomology can be q-deformed to a filtration on q-de Rham cohomology. While such a q-deformed filtration (provably) doesn’t exist in general, I’ll explain how to construct a filtration (along with the “Habiro cohomology”) in many interesting cases, using topological Hochschild homology over the complex K-theory spectrum ku. I’ll also explain how a refined version of THH can be used to obtain a completely functorial construction for varieties over Q.

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