### A Spectral Sequence for Motivic Cohomology, by Spencer Bloch and Steve Lichtenbaum

The purpose of this paper is to construct a spectral sequence
from the motivic cohomology of a field F to its algebraic K-theory:
Here we define motivic cohomology via higher Chow groups.

We admit to a certain presumption in defining motivic cohomology via higher
Chow groups. At this point, one can only say that the higher Chow groups
have some of the expected properties. Those who search for unicorns must
beware being led astray by cows with one horn.

In section 1 we use multi-relative K-theory with supports to define a graded
complex of the form

... --> D --> D --> E --> D --> D --> E --> ...

We show that, assuming a rather innocuous looking "moving lemma" called Theorem
A, this complex is an exact couple, and the resulting spectral sequence has
the desired form. Sections 2 through 6 of the paper are devoted to proving
theorem A.

Finally, in section 7, we prove that the higher Chow complex shifted to the
right 4 steps and then truncated so the resulting complex is supported in
degrees 1 and 2, is quasi-isomorphic to the complex Gamma(2) introduced by
Lichtenbaum. It would follow from a variant of the Soulé Beilinson
conjecture that truncation was unnecessary in this context. Indeed, one
might formulate a "CEO" (cock-eyed optimist) conjecture that Z^r(X,.)[-2r],
sheafified for the étale topology on X, satisfies the six axioms listed
in Lichtenbaum's paper "Values of zeta-functions at non-negative integers"
and that the hypercohomology of this complex is linked to the value of the
Zeta function of X at the point s=r as suggested there.

Spencer Bloch <bloch@math.uchicago.edu>

Steve Lichtenbaum <slicht@brownvm.brown.edu>