Torsion in the 0-cycle group with modulus

We show, for a smooth projective variety $X$ over an algebraically closed field $k$ with an effective Cartier divisor $D$, that the torsion subgroup ${\text{CH}}_0(X\mathrm{\mid }D)\{l\}$ can be described in terms of a relative {'e}tale cohomology for any prime $l\mathrm{\ne }p=\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(k)$. 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 ${\text{CH}}_0(X\mathrm{\mid }D)$ when $D$ is reduced. We deduce applications to the problem of invariance of the prime-to-$p$ torsion in ${\text{CH}}_0(X\mathrm{\mid }D)$ under an infinitesimal extension of $D$.