#### Vol. 12, No. 6, 2018

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Torsion in the 0-cycle group with modulus

### Amalendu Krishna

Vol. 12 (2018), No. 6, 1431–1469
##### 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 $D$.

##### Keywords
Cycles with modulus, cycles on singular schemes, algebraic K-theory, étale cohomology
##### Mathematical Subject Classification 2010
Primary: 14C25
Secondary: 13F35, 14F30, 19F15