We study properties of Bloch’s higher Chow groups on smooth varieties over Dedekind rings. We prove the vanishing of for i > n, and the existence of a Gersten resolution for if the residue characteristic is p. We also show that the Bloch-Kato conjecture implies the Beilinson-Lichtenbaum conjecture an identification for m invertible, and a Gersten resolution with (arbitrary) finite coefficients. Over a complete 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.