Zero-cycle groups on algebraic varieties

Federico Binda; Amalendu Krishna

doi:10.5802/jep.183

Federico Binda ¹; Amalendu Krishna ²

¹ Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano Via Cesare Saldini 50, 20133 Milano, Italy
² Department of Mathematics, Indian Institute of Science Bangalore, 560012, India

Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 281-325.

Abstract
Résumé

We compare various groups of $0$ -cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of $0$ -cycles coincides with the corresponding Friedlander-Voevodsky motivic cohomology. We also show that over an algebraically closed field of positive characteristic, the Chow group of $0$ -cycles with modulus on a smooth projective variety with respect to a reduced divisor coincides with the Suslin homology of the complement of the divisor. We prove several generalizations of the finiteness theorem of Saito and Sato for the Chow group of $0$ -cycles over $p$ -adic fields. We also use these results to deduce a torsion theorem for Suslin homology which extends a result of Bloch to open varieties.

Nous comparons divers groupes de $0$ -cycles sur des variétés quasi-projectives sur un corps. Comme applications, nous montrons que pour certaines variétés projectives singulières, le groupe de Chow de Levine-Weibel des $0$ -cycles coïncide avec la cohomologie motivique correspondante de Friedlander-Voevodsky. Nous montrons également que sur un corps algébriquement clos de caractéristique positive, le groupe de Chow des $0$ -cycles avec modulus sur une variété projective lisse par rapport à un diviseur réduit coïncide avec l’homologie de Suslin du complémentaire du diviseur. Nous prouvons plusieurs généralisations du théorème de finitude de Saito et Sato pour le groupe de Chow des $0$ -cycles sur les corps $p$ -adiques. Nous utilisons également ces résultats pour déduire un théorème de torsion pour l’homologie de Suslin qui étend un résultat de Bloch aux variétés ouvertes.

Received: 2021-04-27
Accepted: 2022-01-10
Published online: 2022-01-20

DOI: 10.5802/jep.183

Classification: 14C25, 14F42, 19E15
Keywords: Cycles with modulus, cycles on singular varieties, motivic cohomology

Author's affiliations:

Federico Binda ¹; Amalendu Krishna ²

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Federico Binda and Amalendu Krishna},
     title = {Zero-cycle groups on algebraic varieties},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {281--325},
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Federico Binda; Amalendu Krishna. Zero-cycle groups on algebraic varieties. Journal de l’École polytechnique — Mathématiques, Volume 9 (2022), pp. 281-325. doi : 10.5802/jep.183. https://jep.centre-mersenne.org/articles/10.5802/jep.183/

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