Torsion in the 0-cycle group with modulus

TL;DR

This paper extends classical results on torsion in Chow groups to the setting with modulus, establishing a description via étale cohomology and proving a Roitman torsion theorem for the zero-cycle group with modulus.

Contribution

It introduces a new description of torsion in zero-cycle groups with modulus using relative étale cohomology and proves the Roitman torsion theorem in this context.

Findings

Torsion subgroup of zero-cycle group with modulus described via étale cohomology.

Proved Roitman torsion theorem including p-torsion for zero-cycle groups with modulus.

Demonstrated invariance of prime-to-p torsion under infinitesimal extensions of the divisor D.

Abstract

We show, for a smooth projective variety $X$ over an algebraically closed field $k$ with an effective Cartier divisor $D$, that the torsion subgroup $\CH_0(X|D)\{l\}$ can be described in terms of a relative {\'e}tale cohomology for any prime $l \neq p = {\rm char}(k)$. This extends a classical result of Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We prove the Roitman torsion theorem (including $p$-torsion) for $\CH_0(X|D)$ when $D$ is reduced. We deduce applications to the problem of invariance of the prime-to-$p$ torsion in $\CH_0(X|D)$ under an infinitesimal extension of $D$.