Regular Morphisms and Gersten's Conjecture

TL;DR

This paper proves that regular morphisms induce acyclic Gersten complexes in a Nisnevich-local setting and reduces the Gersten conjecture to a case involving smooth discrete valuation rings, with applications to algebraic cycles.

Contribution

It establishes a Nisnevich-local acyclicity result for Gersten complexes under regular morphisms and reduces the Gersten conjecture to a simpler case, advancing understanding in algebraic K-theory.

Findings

Gersten complex becomes acyclic in degrees beyond the dimension of Y

Reduction of the Gersten conjecture to smooth discrete valuation rings

Applications to Bloch's Formula and Chow group vanishing

Abstract

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.