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

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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/

References
Cited by

[1] R. Akhtar - “Zero-cycles on varieties over finite fields”, Comm. Algebra 32 (2004) no. 1, p. 279-294 | Article | MR: 2036237 | Zbl: 1062.14013

[2] L. Barbieri-Viale, C. Pedrini & C. Weibel - “Roitman’s theorem for singular complex projective surfaces”, Duke Math. J. 84 (1996) no. 1, p. 155-190 | Article | MR: 1394751 | Zbl: 0864.14026

[3] F. Binda & A. Krishna - “Zero cycles with modulus and zero cycles on singular varieties”, Compositio Math. 154 (2018) no. 1, p. 120-187 | Article | MR: 3719246 | Zbl: 1412.14007

[4] F. Binda & A. Krishna - “Rigidity for relative 0-cycles”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 22 (2021) no. 1, p. 241-267 | MR: 4288654 | Zbl: 1467.14020

[5] F. Binda, A. Krishna & S. Saito - “Bloch’s formula for $0$ -cycles with modulus and higher dimensional class field theory”, J. Algebraic Geom. (to appear), arXiv:2002.01856

[6] F. Binda & S. Saito - “Relative cycles with moduli and regulator maps”, J. Inst. Math. Jussieu 18 (2019) no. 6, p. 1233-1293 | Article | MR: 4021105 | Zbl: 1439.14038

[7] S. Bloch - “Torsion algebraic cycles and a theorem of Roitman”, Compositio Math. 39 (1979) no. 1, p. 107-127 | Numdam | MR: 539002 | Zbl: 0463.14002

[8] D.-C. Cisinski & F. Déglise - “Integral mixed motives in equal characteristic”, Doc. Math. (2015), p. 145-194, Extra vol.: Alexander S. Merkurjev’s sixtieth birthday | MR: 3404379 | Zbl: 1357.19004

[9] H. Esnault & O. Wittenberg - “On the cycle class map for zero-cycles over local fields”, Ann. Sci. École Norm. Sup. (4) 49 (2016) no. 2, p. 483-520, With an appendix by Spencer Bloch | Article | MR: 3481356 | Zbl: 1408.14015

[10] P. Forré - On the kernel of the reciprocity map for varieties over local fields, Ph. D. Thesis, Universität Regensburg, 2011

[11] E. M. Friedlander & V. Voevodsky - “Bivariant cycle cohomology”, in Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton University Press, Princeton, NJ, 2000, p. 138-187 | MR: 1764201 | Zbl: 1019.14011

[12] W. Fulton - Intersection theory, Ergeb. Math. Grenzgeb. (3), vol. 2, Springer-Verlag, Berlin, 1998 | Article

[13] T. Geisser & M. Levine - “The $K$ -theory of fields in characteristic $p$ ”, Invent. Math. 139 (2000) no. 3, p. 459-493 | Article | MR: 1738056 | Zbl: 0957.19003

[14] S. C. Geller & C. A. Weibel - “ $K_{1} (A, B, I)$ ”, J. reine angew. Math. 342 (1983), p. 12-34 | Article | Zbl: 0503.18009

[15] M. Ghosh & A. Krishna - “Bertini theorems revisited”, 2020 | 1912.09076v2

[16] M. Ghosh & A. Krishna - “Zero-cycles on normal varieties”, 2021 | 2012.11249v2

[17] M. Gros & N. Suwa - “Application d’Abel-Jacobi $p$ -adique et cycles algébriques”, Duke Math. J. 57 (1988) no. 2, p. 579-613 | Article | Zbl: 0697.14005

[18] R. Gupta & A. Krishna - “Reciprocity for Kato-Saito idele class group with modulus”, 2020 | 2008.05719v1

[19] R. Gupta & A. Krishna - “Idele class groups with modulus”, 2021 | 2101.04609v1

[20] R. Gupta, A. Krishna & J. Rathore - “A decomposition theorem for $0$ -cycles and applications to class field theory”, 2021 | 2109.10037v1

[21] R. Hartshorne - Algebraic geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, 1977 | Article

[22] A. J. de Jong - “Smoothness, semi-stability and alterations”, Publ. Math. Inst. Hautes Études Sci. (1996) no. 83, p. 51-93 | Article | Numdam | MR: 1423020 | Zbl: 0916.14005

[23] J.-P. Jouanolou - Théorèmes de Bertini et applications, Progress in Math., vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 | MR: 725671

[24] K. Kato & S. Saito - “Unramified class field theory of arithmetical surfaces”, Ann. of Math. (2) 118 (1983) no. 2, p. 241-275 | Article | MR: 717824 | Zbl: 0562.14011

[25] K. Kato & S. Saito - “Global class field theory of arithmetic schemes”, in Applications of algebraic $K$ -theory to algebraic geometry and number theory (Boulder, Colo., 1983), Contemp. Math., vol. 55, American Mathematical Society, Providence, RI, 1986, p. 255-331 | Article | MR: 862639 | Zbl: 0614.14001

[26] S. Kelly - Voevodsky motives and $l$ dh-descent, Astérisque, vol. 391, Société Mathématique de France, Paris, 2017

[27] M. Kerz - “Milnor $K$ -theory of local rings with finite residue fields”, J. Algebraic Geom. 19 (2010) no. 1, p. 173-191 | Article | MR: 2551760 | Zbl: 1190.14021

[28] M. Kerz, H. Esnault & O. Wittenberg - “A restriction isomorphism for cycles of relative dimension zero”, Camb. J. Math. 4 (2016) no. 2, p. 163-196 | Article | MR: 3529393 | Zbl: 1376.14008

[29] M. Kerz & S. Saito - “Chow group of 0-cycles with modulus and higher-dimensional class field theory”, Duke Math. J. 165 (2016) no. 15, p. 2811-2897 | Article | MR: 3557274 | Zbl: 1401.14148

[30] F. Keune - “The relativization of $K_{2}$ ”, J. Algebra 54 (1978) no. 1, p. 159-177 | Article | MR: 511460 | Zbl: 0403.18009

[31] S. L. Kleiman & A. B. Altman - “Bertini theorems for hypersurface sections containing a subscheme”, Comm. Algebra 7 (1979) no. 8, p. 775-790 | Article | MR: 529493 | Zbl: 0401.14002

[32] A. Krishna - “On 0-cycles with modulus”, Algebra Number Theory 9 (2015) no. 10, p. 2397-2415 | Article | MR: 3437766 | Zbl: 1356.14010

[33] A. Krishna - “Torsion in the 0-cycle group with modulus”, Algebra Number Theory 12 (2018) no. 6, p. 1431-1469 | Article | MR: 3864203 | Zbl: 1409.14018

[34] A. Krishna & J. Park - “A module structure and a vanishing theorem for cycles with modulus”, Math. Res. Lett. 24 (2017) no. 4, p. 1147-1176 | Article | MR: 3723807 | Zbl: 1401.14031

[35] A. Krishna & P. Pelaez - “The slice spectral sequence for singular schemes and applications”, Ann. $K$ -Theory 3 (2018) no. 4, p. 657-708 | Article | MR: 3892963 | Zbl: 1423.14156

[36] A. Krishna & P. Pelaez - “Motivic spectral sequence for relative homotopy $K$ -theory”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 21 (2020), p. 411-447 | MR: 4288608 | Zbl: 07373222

[37] M. Levine - “Bloch’s formula for singular surfaces”, Topology 24 (1985) no. 2, p. 165-174 | Article | MR: 793182 | Zbl: 0598.14009

[38] M. Levine - “Torsion zero-cycles on singular varieties”, Amer. J. Math. 107 (1985) no. 3, p. 737-757 | Article | MR: 789661 | Zbl: 0579.14007

[39] M. Levine - “Zero-cycles and $K$ -theory on singular varieties”, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, American Mathematical Society, Providence, RI, 1987, p. 451-462 | MR: 927992 | Zbl: 0635.14007

[40] M. Levine & C. Weibel - “Zero cycles and complete intersections on singular varieties”, J. reine angew. Math. 359 (1985), p. 106-120 | MR: 794801 | Zbl: 0555.14004

[41] C. Mazza, V. Voevodsky & C. Weibel - Lecture notes on motivic cohomology, Clay Math. Monographs, vol. 2, American Mathematical Society, Providence, RI, 2006

[42] J. S. Milne - Étale cohomology, Princeton Math. Series, vol. 33, Princeton University Press, Princeton, NJ, 1980 | MR: 559531

[43] H. Miyazaki - “Cube invariance of higher Chow groups with modulus”, J. Algebraic Geom. 28 (2019) no. 2, p. 339-390 | Article | MR: 3912061 | Zbl: 1454.14022

[44] M. Raynaud & L. Gruson - “Critères de platitude et de projectivité. Techniques de “platification” d’un module”, Invent. Math. 13 (1971), p. 1-89 | Article | Zbl: 0227.14010

[45] H. Russell - “Albanese varieties with modulus over a perfect field”, Algebra Number Theory 7 (2013) no. 4, p. 853-892 | Article | MR: 3095229 | Zbl: 1282.14078

[46] S. Saito & K. Sato - “A finiteness theorem for zero-cycles over $p$ -adic fields”, Ann. of Math. (2) 172 (2010) no. 3, p. 1593-1639, With an appendix by Uwe Jannsen | Article | MR: 2726095 | Zbl: 1210.14012

[47] A. Schmidt - “Singular homology of arithmetic schemes”, Algebra Number Theory 1 (2007) no. 2, p. 183-222 | Article | MR: 2361940 | Zbl: 1184.19002

[48] J.-P. Serre - “Morphismes universels et différentielles de troisième espèce”, in Variétés de Picard, Séminaire C. Chevalley (1958/59), vol. 3, Secrétariat mathématique, Paris, 1960, Exp. no. 11 | Zbl: 0123.14001

[49] M. Spieß & T. Szamuely - “On the Albanese map for smooth quasi-projective varieties”, Math. Ann. 325 (2003) no. 1, p. 1-17 | Article | MR: 1957261 | Zbl: 1077.14026

[50] M. Temkin - “Tame distillation and desingularization by $p$ -alterations”, Ann. of Math. (2) 186 (2017) no. 1, p. 97-126 | Article | MR: 3665001 | Zbl: 1370.14015

[51] R. W. Thomason & T. Trobaugh - “Higher algebraic $K$ -theory of schemes and of derived categories”, in The Grothendieck Festschrift, Vol. III, Progress in Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, p. 247-435 | Article | MR: 1106918

[52] V. Voevodsky - “Triangulated categories of motives over a field”, in Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., vol. 143, Princeton University Press, Princeton, NJ, 2000, p. 188-238 | MR: 1764202 | Zbl: 1019.14009

[53] V. Voevodsky - “Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic”, Internat. Math. Res. Notices (2002) no. 7, p. 351-355 | Article | MR: 1883180 | Zbl: 1057.14026

[54] C. A. Weibel - “Homotopy algebraic $K$ -theory”, in Algebraic $K$ -theory and algebraic number theory (Honolulu, HI, 1987), Contemp. Math., vol. 83, American Mathematical Society, Providence, RI, 1989, p. 461-488 | Article | MR: 991991 | Zbl: 0669.18007

[55] O. Zariski - Introduction to the problem of minimal models in the theory of algebraic surfaces, Publ. Math. Soc. Japan, vol. 4, The Mathematical Society of Japan, Tokyo, 1958 | MR: 97403

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