  We prove that if $X \to Y$ 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 $\mathbb{Z}$. 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.
