Topological K-theory of Algebraic K-theory Spectra

One of the central problems of algebraic K-theory is to compute the K-groups KX of a scheme X. Since these groups are, by definition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopy-type of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the K-groups themselves, knowledge of the spectrum KX can be applied to the study of other topological invariants. For example, if X = SpecR, then the homology groups of the zero-th space ΩKX are of interest since they are the homology groups of the infinite general linear group GLR; but they are not determined by the homotopy groups of KX alone. Topological complex K-theory is another important invariant. Let K denote the periodic complex K-theory spectrum, and let K̂ denote its Bousfield `-adic completion at a fixed prime `. There are several reasons for wanting to compute K̂∗KX. From the standpoint of stable homotopy theory, (K̂/`)∗ is essentially the second in an infinite hierarchy of cohomology theories—the Morava K-theories: K(0), K(1), K(2)... Here K(0) is rational cohomology. By work of Hopkins and J. Smith [10], these theories are “primes” of the `-local stable homotopy category, and together with mod ` cohomology as a sort of “prime at infinity” they form a complete list of the distinct primes. Moreover, by a theorem of the author [13], all the Morava K-theories beyond (K̂/`)∗ vanish on KX. Therefore K̂∗ has a distinguished role in the theory. Even better, K̂∗KX is accessible to computation, thanks to Thomason’s theorem verifying the Lichtenbaum-Quillen conjectures for L̂KX, the Bousfield localization with respect to K̂. The author and Bill Dwyer explicitly computed K̂∗KX when X is a ring of integers in a number field [7], a smooth curve over a finite field [8], or a local field [7] (with some exceptions at the prime 2 that were left unresolved; see below for further discussion). In these cases the classical Iwasawa theory of X is very neatly captured by the structure of K̂−1KX as module over the Adams operations. The present paper provides a more systematic, general approach to computing K̂∗KX. As in [7], Thomason’s theorem is still a critical input, and therefore our results apply only to schemes satisfying the Thomason hypotheses (T) listed below. In particular X is a separated noetherian regular scheme of finite Krull dimension, the prime ` is invertible in X, and X