Some Conjectures on the Algebraic K-Theory of Fields, I: K-Theory with Coefficients and Étale K-Theory

Abstract

We investigate relationships between K-theory with coefficients and étale cohomology. Classically, such relationships are given by a) homomorphisms from the former towards the latter (Chern classes) and b) comparison with étale K-theory, which is the abutment of a spectral sequence starting with étale cohomology groups. The main theme of this paper is to explain that, locally for the Zariski topology, there should be homomorphisms in the opposite direction to a), and that these homomorphisms should split the étale K-theory spectral sequence in a very strong sense. As a consequence, étale K-theory groups of a nice semi-local ring should be isomorphic to a direct sum of étale cohomology groups, and a part of this sum should map to ordinary K-groups (with coefficients) as a direct summand. We further conjecture that these split injections should actually be isomorphisms: this is equivalent to conjecturing that ordinary K-theory injects into étale K-theory, or that Bott elements are nonzero divisors in the former. Given the conjectural homomorphisms above, this would also be a formal consequence of the existence of a spectral sequence from étale cohomology to ordinary K-theory, as conjectured by Beilinson.

We actually construct some of these “anti-Chern classes” in lower cohomology dimension, and indicate how one should be able to construct higher ones. Among other things, the construction rests on the MIlnor-Kato conjecture relating Milnor’s K-theory to étale cohomology. Various applications are given.