Nous comparons divers groupes de -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 -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 -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 -cycles sur les corps -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.
We compare various groups of -cycles on quasi-projective varieties over a field. As applications, we show that for certain singular projective varieties, the Levine-Weibel Chow group of -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 -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 -cycles over -adic fields. We also use these results to deduce a torsion theorem for Suslin homology which extends a result of Bloch to open varieties.
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Keywords: Cycles with modulus, cycles on singular varieties, motivic cohomology
Mot clés : Cycles avec modulus, cycles sur les variétés singulières, cohomologie motivique
@article{JEP_2022__9__281_0, author = {Binda, Federico and Krishna, Amalendu}, title = {Zero-cycle groups on algebraic varieties}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {281--325}, publisher = {Ecole polytechnique}, volume = {9}, year = {2022}, doi = {10.5802/jep.183}, language = {en}, url = {http://www.numdam.org/articles/10.5802/jep.183/} }
TY - JOUR AU - Binda, Federico AU - Krishna, Amalendu TI - Zero-cycle groups on algebraic varieties JO - Journal de l’École polytechnique — Mathématiques PY - 2022 SP - 281 EP - 325 VL - 9 PB - Ecole polytechnique UR - http://www.numdam.org/articles/10.5802/jep.183/ DO - 10.5802/jep.183 LA - en ID - JEP_2022__9__281_0 ER -
Binda, Federico; Krishna, Amalendu. Zero-cycle groups on algebraic varieties. Journal de l’École polytechnique — Mathématiques, Tome 9 (2022), pp. 281-325. doi : 10.5802/jep.183. http://www.numdam.org/articles/10.5802/jep.183/
[1] Zero-cycles on varieties over finite fields, Comm. Algebra, Volume 32 (2004) no. 1, pp. 279-294 | DOI | MR | Zbl
[2] Roitman’s theorem for singular complex projective surfaces, Duke Math. J., Volume 84 (1996) no. 1, pp. 155-190 | DOI | MR | Zbl
[3] Zero cycles with modulus and zero cycles on singular varieties, Compositio Math., Volume 154 (2018) no. 1, pp. 120-187 | DOI | MR | Zbl
[4] Rigidity for relative 0-cycles, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 22 (2021) no. 1, pp. 241-267 | MR | Zbl
[5] Bloch’s formula for -cycles with modulus and higher dimensional class field theory, J. Algebraic Geom. (to appear) (arXiv:2002.01856)
[6] Relative cycles with moduli and regulator maps, J. Inst. Math. Jussieu, Volume 18 (2019) no. 6, pp. 1233-1293 | DOI | MR | Zbl
[7] Torsion algebraic cycles and a theorem of Roitman, Compositio Math., Volume 39 (1979) no. 1, pp. 107-127 | Numdam | MR | Zbl
[8] Integral mixed motives in equal characteristic, Doc. Math. (2015), pp. 145-194 (Extra vol.: Alexander S. Merkurjev’s sixtieth birthday) | MR | Zbl
[9] On the cycle class map for zero-cycles over local fields, Ann. Sci. École Norm. Sup. (4), Volume 49 (2016) no. 2, pp. 483-520 (With an appendix by Spencer Bloch) | DOI | MR | Zbl
[10] On the kernel of the reciprocity map for varieties over local fields, Ph. D. Thesis, Universität Regensburg (2011)
[11] Bivariant cycle cohomology, Cycles, transfers, and motivic homology theories (Ann. of Math. Stud.), Volume 143, Princeton University Press, Princeton, NJ, 2000, pp. 138-187 | MR | Zbl
[12] Intersection theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, Berlin, 1998 | DOI
[13] The -theory of fields in characteristic , Invent. Math., Volume 139 (2000) no. 3, pp. 459-493 | DOI | MR | Zbl
[14] , J. reine angew. Math., Volume 342 (1983), pp. 12-34 | DOI | Zbl
[15] Bertini theorems revisited, 2020 | arXiv
[16] Zero-cycles on normal varieties, 2021 | arXiv
[17] Application d’Abel-Jacobi -adique et cycles algébriques, Duke Math. J., Volume 57 (1988) no. 2, pp. 579-613 | DOI | Zbl
[18] Reciprocity for Kato-Saito idele class group with modulus, 2020 | arXiv
[19] Idele class groups with modulus, 2021 | arXiv
[20] A decomposition theorem for -cycles and applications to class field theory, 2021 | arXiv
[21] Algebraic geometry, Graduate Texts in Math., 52, Springer-Verlag, 1977 | DOI
[22] Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci. (1996) no. 83, pp. 51-93 | DOI | Numdam | MR | Zbl
[23] Théorèmes de Bertini et applications, Progress in Math., 42, Birkhäuser Boston, Inc., Boston, MA, 1983, ii+127 pages | MR
[24] Unramified class field theory of arithmetical surfaces, Ann. of Math. (2), Volume 118 (1983) no. 2, pp. 241-275 | DOI | MR | Zbl
[25] Global class field theory of arithmetic schemes, Applications of algebraic -theory to algebraic geometry and number theory (Boulder, Colo., 1983) (Contemp. Math.), Volume 55, American Mathematical Society, Providence, RI, 1986, pp. 255-331 | DOI | MR | Zbl
[26] Voevodsky motives and dh-descent, Astérisque, 391, Société Mathématique de France, Paris, 2017
[27] Milnor -theory of local rings with finite residue fields, J. Algebraic Geom., Volume 19 (2010) no. 1, pp. 173-191 | DOI | MR | Zbl
[28] A restriction isomorphism for cycles of relative dimension zero, Camb. J. Math., Volume 4 (2016) no. 2, pp. 163-196 | DOI | MR | Zbl
[29] Chow group of 0-cycles with modulus and higher-dimensional class field theory, Duke Math. J., Volume 165 (2016) no. 15, pp. 2811-2897 | DOI | MR | Zbl
[30] The relativization of , J. Algebra, Volume 54 (1978) no. 1, pp. 159-177 | DOI | MR | Zbl
[31] Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra, Volume 7 (1979) no. 8, pp. 775-790 | DOI | MR | Zbl
[32] On 0-cycles with modulus, Algebra Number Theory, Volume 9 (2015) no. 10, pp. 2397-2415 | DOI | MR | Zbl
[33] Torsion in the 0-cycle group with modulus, Algebra Number Theory, Volume 12 (2018) no. 6, pp. 1431-1469 | DOI | MR | Zbl
[34] A module structure and a vanishing theorem for cycles with modulus, Math. Res. Lett., Volume 24 (2017) no. 4, pp. 1147-1176 | DOI | MR | Zbl
[35] The slice spectral sequence for singular schemes and applications, Ann. -Theory, Volume 3 (2018) no. 4, pp. 657-708 | DOI | MR | Zbl
[36] Motivic spectral sequence for relative homotopy -theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume 21 (2020), pp. 411-447 | MR | Zbl
[37] Bloch’s formula for singular surfaces, Topology, Volume 24 (1985) no. 2, pp. 165-174 | DOI | MR | Zbl
[38] Torsion zero-cycles on singular varieties, Amer. J. Math., Volume 107 (1985) no. 3, pp. 737-757 | DOI | MR | Zbl
[39] Zero-cycles and -theory on singular varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Proc. Sympos. Pure Math.), Volume 46, American Mathematical Society, Providence, RI, 1987, pp. 451-462 | MR | Zbl
[40] Zero cycles and complete intersections on singular varieties, J. reine angew. Math., Volume 359 (1985), pp. 106-120 | MR | Zbl
[41] Lecture notes on motivic cohomology, Clay Math. Monographs, 2, American Mathematical Society, Providence, RI, 2006
[42] Étale cohomology, Princeton Math. Series, 33, Princeton University Press, Princeton, NJ, 1980 | MR
[43] Cube invariance of higher Chow groups with modulus, J. Algebraic Geom., Volume 28 (2019) no. 2, pp. 339-390 | DOI | MR | Zbl
[44] Critères de platitude et de projectivité. Techniques de “platification” d’un module, Invent. Math., Volume 13 (1971), pp. 1-89 | DOI | Zbl
[45] Albanese varieties with modulus over a perfect field, Algebra Number Theory, Volume 7 (2013) no. 4, pp. 853-892 | DOI | MR | Zbl
[46] A finiteness theorem for zero-cycles over -adic fields, Ann. of Math. (2), Volume 172 (2010) no. 3, pp. 1593-1639 (With an appendix by Uwe Jannsen) | DOI | MR | Zbl
[47] Singular homology of arithmetic schemes, Algebra Number Theory, Volume 1 (2007) no. 2, pp. 183-222 | DOI | MR | Zbl
[48] Morphismes universels et différentielles de troisième espèce, Variétés de Picard (Séminaire C. Chevalley (1958/59)), Volume 3, Secrétariat mathématique, Paris, 1960 (Exp. no. 11) | Zbl
[49] On the Albanese map for smooth quasi-projective varieties, Math. Ann., Volume 325 (2003) no. 1, pp. 1-17 | DOI | MR | Zbl
[50] Tame distillation and desingularization by -alterations, Ann. of Math. (2), Volume 186 (2017) no. 1, pp. 97-126 | DOI | MR | Zbl
[51] Higher algebraic -theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III (Progress in Math.), Volume 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247-435 | DOI | MR
[52] Triangulated categories of motives over a field, Cycles, transfers, and motivic homology theories (Ann. of Math. Stud.), Volume 143, Princeton University Press, Princeton, NJ, 2000, pp. 188-238 | MR | Zbl
[53] Motivic cohomology groups are isomorphic to higher Chow groups in any characteristic, Internat. Math. Res. Notices (2002) no. 7, pp. 351-355 | DOI | MR | Zbl
[54] Homotopy algebraic -theory, Algebraic -theory and algebraic number theory (Honolulu, HI, 1987) (Contemp. Math.), Volume 83, American Mathematical Society, Providence, RI, 1989, pp. 461-488 | DOI | MR | Zbl
[55] Introduction to the problem of minimal models in the theory of algebraic surfaces, Publ. Math. Soc. Japan, 4, The Mathematical Society of Japan, Tokyo, 1958 | MR
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