Hopf algebra structure on topological Hochschild homology

Abstract

The topological Hochschild homology $THH\left(R\right)$ of a commutative $S$–algebra ($E_{\infty }$ ring spectrum) $R$ naturally has the structure of a commutative $R$–algebra in the strict sense, and of a Hopf algebra over $R$ in the homotopy category. We show, under a flatness assumption, that this makes the Bökstedt spectral sequence converging to the mod $p$ homology of $THH\left(R\right)$ into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative $S$–algebras $ku$, $ko$, $tmf$, $ju$ and $j$, and to calculate the homotopy groups of $THH\left(ku\right)$ and $THH\left(ko\right)$ after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic $K$–theory of $S$–algebras, by means of the cyclotomic trace map to topological cyclic homology.