Higher topological Hochschild homology of periodic complex K-theory

Abstract

We describe the topological Hochschild homology of the periodic complex $K$-theory spectrum, $THH(KU)$, as a commutative $KU$-algebra: it is equivalent to $KU[K(\mathbb{Z},3)]$ and to $F(\Sigma K{U}_{\mathbb{Q}})$, where $F$ is the free commutative $KU$-algebra functor on a $KU$-module. Moreover, $F(\Sigma K{U}_{\mathbb{Q}})\simeq KU\vee \Sigma K{U}_{\mathbb{Q}}$, a square-zero extension. In order to prove these results, we first establish that topological Hochschild homology commutes, as an algebra, with localization at an element. Then, we prove that $TH{H}^n(KU)$, the $n$-fold iteration of $THH(KU)$, i.e. $T^n\otimes KU$, is equivalent to $KU[G]$ where $G$ is a certain product of integral Eilenberg-Mac Lane spaces, and to a free commutative $KU$-algebra on a rational $KU$-module. We prove that $S^n\otimes KU$ is equivalent to $KU[K(\mathbb{Z},n+2)]$ and to $F({\Sigma }^{n}K{U}_{\mathbb{Q}})$. We describe the topological André-Quillen homology of $KU$.

arXiv comment40 pages. Final version