Comparing K-theories for complex varieties, by Eric M. Friedlander and Mark E. Walker

The semi-topological K-theory of a complex variety was defined previously by the authors with the expectation that it would prove to be a theory lying ``part way'' between the algebraic K-theory of the variety and the topological K-theory of the associated analytic space, and thus would share properties with each of these other theories. In this paper, we realize these expectations by proving among other results that (1) the algebraic K-theory with finite coefficients and the semi-topological K-theory with finite coefficients coincide on all projective complex varieties, (2) semi-topological K-theory and topological K-theory agree on certain types of generalized flag varieties, and (3) (by building on a result of Cohen and Lima-Filho) the semi-topological K-theory of any smooth variety becomes isomorphic to the topological K-theory of the underlying analytic space once the Bott element is inverted. To illustrate the utility of our results, we observe that a new proof of the Quillen-Lichtenbaum conjecture for smooth, complete curves is obtained as a corollary.

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