Torsion zero cycles with modulus on affine varieties

TL;DR

This paper proves that the Chow group of zero cycles with modulus on smooth affine varieties over algebraically closed fields is torsion free except possibly for p-torsion in positive characteristic, generalizing classical theorems.

Contribution

It extends classical results of Rojtman and Levine to the setting of zero cycles with modulus on affine varieties, showing torsion freeness except for p-torsion.

Findings

Chow group ${ m CH}_0(X|D)$ is torsion free in characteristic zero.

In positive characteristic, torsion may only be p-torsion.

Generalizes classical theorems to the modulus setting.

Abstract

In this note we show that given a smooth affine variety $X$ over an algebraically closed field $k$ and an effective (possibly non reduced) Cartier divisor $D$ on it, the Kerz-Saito Chow group of zero cycles with modulus ${\rm CH}_0(X|D)$ is torsion free, except possibly for $p$-torsion if the characteristic of $k$ is $p>0$. This generalizes to the relative setting classical theorems of Rojtman (for $X$ smooth) and of Levine (for $X$ singular). A stronger version of this result, that encompasses $p$-torsion as well, was proven with a different and more sophisticated method by A. Krishna and the author in another paper.