Rigidity for relative $0$-cycles

TL;DR

This paper establishes an isomorphism between the classical Chow group of relative 0-cycles on certain schemes and the Levine-Weibel Chow group on the special fiber, under specific conditions, extending previous results.

Contribution

It generalizes the relation between classical and Levine-Weibel Chow groups for relative 0-cycles on regular schemes over Henselian DVRs, with finite coefficients.

Findings

Classical and Levine-Weibel Chow groups are isomorphic under certain conditions.

The result extends previous work by Esnault, Kerz, and Wittenberg.

The isomorphism holds with finite coefficients.

Abstract

We present a relation between the classical Chow group of relative $0$-cycles on a regular scheme $\mathcal{X}$, projective and flat over an excellent Henselian discrete valuation ring, and the Levine-Weibel Chow group of 0-cycles on the special fiber. We show that these two Chow groups are isomorphic with finite coefficients under extra assumptions. This generalizes a result of Esnault, Kerz and Wittenberg.