0-cycles on singular schemes and class field theory

TL;DR

This paper establishes a connection between 0-cycle Chow groups on singular schemes over finite fields and class field theory, providing new proofs and extending known formulas to broader contexts.

Contribution

It introduces a new description of abelian extensions via 0-cycle Chow groups on singular schemes and proves the Bloch-Quillen formula in this setting.

Findings

Chow group of 0-cycles characterizes unramified abelian extensions.

Bloch-Quillen formula holds for singular schemes over finite fields.

Simplified proofs of Kerz-Saito results for certain surfaces.

Abstract

We show that the Chow group of 0-cycles on a singular projective scheme $X$ over a finite field describes the abelian extensions of its function field which are unramified over the regular locus of $X$. As a consequence, we obtain the Bloch-Quillen formula for the Chow group of 0-cycles on such schemes. We deduce simple proofs of results of Kerz-Saito for a class of surfaces without any assumption on ${\rm char}(k)$.