Algebraic K-theory of topological K-theory

Abstract

We are interested in the arithmetic of ring spectra. To make sense of this we must work with structured ring spectra, such as S-algebras [EKMM], symmetric ring spectra [HSS] or Γ-rings [Ly]. We will refer to these as Salgebras. The commutative objects are then commutative S-algebras. The category of rings is embedded in the category of S-algebras by the Eilenberg– MacLane functor R →HR. We may therefore view an S-algebra as a generalization of a ring in the algebraic sense. The added flexibility of S-algebras provides room for new examples and constructions, which may eventually also shed light upon the category of rings itself. In algebraic number theory the arithmetic of the ring of integers in a number field is largely captured by its Picard group, its unit group and its Brauer group. These are