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.

中文翻译：

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THH 和 TC（非常）远不是同伦函子

我们计算方案的几个不变量的 $\mathbb{A}^1$-localization，即拓扑 Hochschild homology ($\mathrm{THH}$)、拓扑循环同调 ($\mathrm{TC}$) 和拓扑周期循​​环同源性（$\mathrm{TP}$）。这个过程非常残酷，并且会杀死大多数这些不变量的完整版本。消失陈述的主要成分是 $\mathbb{A}^1$ - de Rham 上同调（以及最终结晶上同调）在正特征中的消失。