Higher topological Hochschild homology of periodic complex K-theory

Abstract

We describe the topological Hochschild homology of the periodic complex K-theory spectrum, T HH(KU), as a commutative KU-algebra: it is equivalent to KU [K(Z, 3)] and to F (ΣKU Q), where F is the free commutative KU-algebra functor on a KU-module. Moreover, F (ΣKU Q) KU ∨ ΣKU 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 T HH n (KU), the n-fold iteration of T HH(KU), i.e. T n ⊗ 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 ⊗KU is equivalent to KU [K(Z, n + 2)] and to F (Σ n KU Q). We describe the topological André-Quillen homology of KU as KU Q .

Stability of Loday constructions

“…Note that by Corollary 4.5 spherical abelian monoid rings are not stable in general, whereas spherical abelian group rings are. Bruno Stonek calculates higher THH of periodic complex topological K-theory, KU , and he determines topological André-Quillen homology of KU [24]. He uses Snaith's description of KU as the Bott localization of Σ ∞ + CP ∞ .…”

“…it has been calculated in many cases. Higher order topological Hochschild homology, which is L R S n (A; C), has also been determined in many important classes of examples, see for instance [3,8,13,22,24]. In [3] we develop several tools for calculating L R ΣX (A; C).…”

The cotangent complex and Thom spectra

“…This allows us, for example, to recover the equivalence L K U K U ∧ H Q from [24], where K U denotes the E ∞ -ring spectrum of periodic complex topological K -theory.…”