Finiteness results for certain motivic cohomology groups
by
Uwe Jannsen
University of Regensburg, 93040 Regensburg, Germany
Coauthors: Shuji Saito

This is a report on joint work in progress with Shuji Saito. We prove finiteness for certain motivic cohomology groups modulo n, viz., the groups CH^d(X, 1, Z/n) where X is a regular proper scheme of dimension d over Z and n is an integer. The elements in this group are represented by families of rational functions on curves (geometric or arithmetic) on X for which the sums of their divisors is divisible by n. We have to assume the Milnor-Bloch-Kato conjecture on the bijectivity of the Galois symbol for the Milnor K-group K^M₃(F)/n, i.e., modulo all primes p dividing n. Thus the result is unconditional for n a power of 2. We have a complete result if X is smooth and projective over a finite field. For the arithmetic case, i.e., X flat over Z, we have a complete result for the case where d is at most 3 or if X has good reduction at all primes dividing n.

Date received: April 1, 2005