THH and TC are (very) far from being homotopy functors

Abstract

We compute the ${\mathbb{A}}^1$-localization of several invariants of schemes namely, topological Hochschild homology ($\mathrm{THH}$), topological cyclic homology ($\mathrm{TC}$) and topological periodic cyclic homology ($\mathrm{TP}$). 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 ${\mathbb{A}}^1$-localization of de Rham cohomology (and, eventually, crystalline cohomology) in positive characteristics.

10 pages; final version, to appear in JPAA