Proceedings of Symposia in Pure Mathematics Topological Cyclic Homology of Schemes

In recent years, the topological cyclic homology functor of [4] has been used to study and to calculate higher algebraic K-theory. It is known that for finite algebras over the ring of Witt vectors of a perfect field of characteristic p, the p-adic K-theory and topological cyclic homology agree in non-negative degrees, [20]. This has been used to calculate the p-adic K-theory of truncated polynomial algebras over perfect fields of characteristic p > 0, [21], and of rings of integers in unramified extensions of the field Qp of p-adic numbers, [6]. In this paper, we extend the definition of topological cyclic homology to schemes. The topological Hochschild spectrum TH(A), recalled in paragraph 2 below, defines, as the ring A varies, a presheaf of spectra on the category of affine schemes. We show in paragraph 3 that the corresponding presheaves of homotopy groups are quasi-coherent sheaves for the étale topology. It follows that the map TH(A) ∼ −→ H((SpecA)ét,TH),

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