Zero-cycle groups on algebraic varieties

TL;DR

This paper investigates various groups of 0-cycles on algebraic varieties, establishing their relationships and equivalences in different contexts, and extends several finiteness and torsion results in algebraic geometry.

Contribution

It demonstrates the equivalence of Levine-Weibel Chow groups with motivic cohomology for certain singular varieties and generalizes finiteness theorems for 0-cycle groups over p-adic fields.

Findings

Levine-Weibel Chow group coincides with motivic cohomology for some singular varieties

Finiteness theorems for Chow groups over p-adic fields are extended

A torsion theorem for Suslin homology is established

Abstract

We compare various groups of 0-cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of 0-cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of 0-cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sato for the Chow group of 0-cycles over $p$-adic fields. We also use these results to deduce a torsion theorem for Suslin homology which extends a result of Bloch to open varieties.