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@math.usc.edu> Lars Hesselholt <larsh@math.mit.edu>