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>