### 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>