The non-nil-invariance of TP

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.