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(Z,3)] and to F(ΣKUQ), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F(ΣKUQ)≃KU∨ΣKUQ, 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 THHn(KU), the n-fold iteration of THH(KU), i.e. Tn⊗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 Sn⊗KU is equivalent to KU[K(Z,n+2)] and to F(ΣnKUQ). We describe the topological André-Quillen homology of KU as KUQ.