### Motivic cohomology over Dedekind rings, by Thomas Geisser

This is a completely rewritten version of Preprint 499.
We study properties of Bloch's higher Chow groups on smooth varieties over
Dedekind rings. We prove a conditional Gersten resolution, which implies that
the motivic complex is acyclic above degree n, and that there is a Gersten
resolution for motivic cohomology with mod p coefficients if the residue
characteristic is p.
We also show that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum
conjecture, and a Gersten resolution with (arbitrary) finite
coefficients. Over a discrete valuation ring of mixed characteristic (0,p),
we construct a map from motivic cohomology to syntomic cohomology, which is a
quasi-isomorphism provided the Bloch-Kato conjecture holds.

Thomas Geisser <geisser@math.usc.edu>