Abstract

Introduction One of the central problems of algebraic K-theory is to compute the K-groups K n X of a scheme X. Since these groups are, by denition, 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 = Spec R, then the homology groups of the zero-th space 1 KX are of interest since they are the homology groups of the innite 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 ^ K denote its Bouseld `-adic completion

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