Abstract
We compute the A 1 -localization of several invariants of schemes namely, topological Hochschild homology ( THH ), topological cyclic homology ( TC ) and topological periodic cyclic homology ( TP ). This procedure is quite brutal and kills the completed versions of most of these invariants. The main ingredient for the vanishing statements is the vanishing of A 1 -localization of de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.
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