### Topological Cyclic Homology of Schemes, by Thomas Geisser and Lars Hesselholt

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.

Thomas Geisser <geisser@exp-math.uni-essen.de>

Lars Hesselholt <lh@math.princeton.edu>