Regular Morphisms and Gersten's Conjecture

Original · EN

We prove that if X → Y is a (geometrically) regular morphism of Noetherian schemes, then from a Nisnevich-local perspective, the Gersten complex for Quillen K-theory on X becomes acyclic in degrees beyond the Krull dimension of Y. Using our methods, we also reduce the general Gersten conjecture for regular, unramified local rings to the case of a discrete valuation ring which is essentially smooth over Z. We apply our results to the the theory of algebraic cycles --- globally to obtain relative versions of Bloch's Formula and locally to address the Claborn-Fossum Conjecture concerning the vanishing of Chow groups for regular local rings.