The non-nil-invariance of TP

TL;DR

This paper demonstrates that the invariance property of periodic topological cyclic homology (TP) under nilpotent extensions, known for Hochschild and cyclic homology, fails in positive characteristic, even rationally.

Contribution

It provides the first example showing that TP is not nil-invariant in positive characteristic, contrasting with known invariance results for HP.

Findings

TP does not preserve invariance under nilpotent extensions in positive characteristic

The failure occurs even after rationalization

Contrasts with the invariance property of Hochschild and cyclic homology

Abstract

Hesselholt defined a spectrum $\operatorname{TP}(X)$, the periodic topological cyclic homology of a scheme $X$, using topological Hochschild homology and the Tate construction, which is a topological analogue of Connes-Tsygan periodic cyclic homology $\operatorname{HP}$ defined by Hochschild homology and the Tate construction. Goodwillie proved that for $R$ an algebra over a field of characteristic 0 and $I$ a nilpotent ideal of $R$, the quotient map $R\to R/I$ induces an isomorphism on $\operatorname{HP}$. In this paper, we show that the analogous result for $\operatorname{TP}$ does not hold, that is to say, there is an algebra of positive characteristic and a nilpotent ideal such that the quotient map does not induce an isomorphism on $\operatorname{TP}$, even rationally.