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 <>