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

We compute the ${\mathbb{A}}^1$-localization of several invariants of schemes namely, topological Hochschild homology ($\mathrm{T}\mathrm{H}\mathrm{H}$), topological cyclic homology ($\mathrm{T}\mathrm{C}$) and topological periodic cyclic homology ($\mathrm{T}\mathrm{P}$). 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.