The non-nil-invariance of TP

Ryo Horiuchi

Published 2017-12-08Version 1

Hesselholt defined a spectrum $\operatorname{TP}(X)$, called periodic topological cyclic homology, for a scheme $X$ using topological Hochshild homology and the Tate construction, which is a topological analogue of the 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 isomorphim on $\rm HP$. In this article, we show that the analogous result for $\rm TP$ does not hold, that is, 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.