Motivic complexes and the K-theory of automorphisms, by Mark E. Walker

In this paper we investigate a proposed definition of the motivic complexes of a regular ring R which involves the K-theory of the category of projective R-modules equipped with commuting automorphisms. In particular, we prove that these complexes are ``correct'' for algebraically closed fields after tensoring with the integers mod n, provided the characteristic of the field does not divide n.

Note added May 23, 2002:

Andrei Suslin has pointed out to me that the proof of Theorem 3.14 has a serious gap and, moreover, that the approach I used to prove this result cannot possibly be made to work. (This preprint has never appeared in print.)

The assertion of Theorem 3.14 is that, for finite coefficients, the motivic cohomology groups of Grayson (defined using direct sum K-groups of automorphisms) coincide with the usual motivic cohomology groups for an algebraically closed field. (By ``usual'', I mean those groups defined and studied by Suslin and Voevodsky and which coincide with Bloch's higher Chow groups for smooth varieties.) Since the posting of this preprint, Tao Wu has proven that, for rational coefficients, Grayson's motivic cohomology groups coincide with the usual ones locally (for the Zariski topology) on a smooth variety. More recently, Andrei Suslin has announced a proof that Grayson's motivic cohomology groups coincide with the usual motivic cohomology groups integrally (locally on smooth varieties). In combination with Grayson's work, Suslin's result establishes a new proof of an Atiyah-Hirzebruch type spectral sequence relating the motivic cohomology groups and the K-groups of the semi-localization of a smooth variety at finite number of points.


Mark E. Walker <mwalker@math.unl.edu>