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

TL;DR

The paper investigates how certain topological invariants of schemes, like THH and TC, behave under $A^1$-localization, revealing they are not homotopy functors and often vanish after localization.

Contribution

It computes the $A^1$-localization of THH, TC, and TP, showing these invariants are not homotopy functors and often vanish, especially in positive characteristic.

Findings

$A^1$-localization kills most invariants' completed versions.

Vanishing of $A^1$-localization of de Rham and crystalline cohomology in positive characteristic.

THH and TC are not homotopy functors after localization.

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.