Topological cyclic homology of schemes (with Thomas Geisser)

We use Thomasons's hypercohomology construction to extend the definition of topological cyclic homology to schemes. We justify this definition by showing that it agrees with the previous definition for affine schemes, and show that it does not depend on the topology coarser than the etale topology.

For smooth schemes over perfect fields of characteristic p we identify the topological cyclic homology sheaf for the Zariski and etale topology; in the etale topology it agrees with the p-completed K-theory sheaf. This is used to relate K-theory and topological cyclic homology in many cases. For example, we calculate topological cyclic homology for any field of characteristic p in terms of K-theory.


Lars Hesselholt <larsh@math.nagoya-u.ac.jp>