Torsion in the 0-cycle group with modulus

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 ${CH}_0\left(X|D\right)\left\{l\right\}$ can be described in terms of a relative étale cohomology for any prime $l\ne p=char\left(k\right)$. 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 ${CH}_0\left(X|D\right)$ when $D$ is reduced. We deduce applications to the problem of invariance of the prime-to-$p$ torsion in ${CH}_0\left(X|D\right)$ under an infinitesimal extension of $D$.