𝐾-theory and 0-cycles on schemes

Gupta, Rahul; Krishna, Amalendu

doi:10.1090/jag/744

$K$-theory and $0$-cycles on schemes

Authors: Rahul Gupta and Amalendu Krishna
Journal: J. Algebraic Geom. 29 (2020), 547-601
DOI: https://doi.org/10.1090/jag/744
Published electronically: November 4, 2019
MathSciNet review: 4158460
Full-text PDF

Abstract | References | Additional Information

Abstract: We prove Bloch’s formula for 0-cycles on affine schemes over algebraically closed fields. We prove this formula also for projective schemes over algebraically closed fields which are regular in codimension one. Several applications, including Bloch’s formula for 0-cycles with modulus, are derived.

References

Steven L. Kleiman and Allen B. Altman, Bertini theorems for hypersurface sections containing a subscheme, Comm. Algebra 7 (1979), no. 8, 775–790. MR 529493, DOI https://doi.org/10.1080/00927877908822375

A. Asok and J. Fasel, Euler class groups and motivic stable cohomotopy, arXiv:1601.05723, 2016.

H. Bass and J. Tate, The Milnor ring of a global field, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972) Springer, Berlin, 1973, pp. 349–446. Lecture Notes in Math., Vol. 342. MR 0442061, DOI https://doi.org/10.1007/BFb0073733

S. M. Bhatwadekar and Raja Sridharan, Zero cycles and the Euler class groups of smooth real affine varieties, Invent. Math. 136 (1999), no. 2, 287–322. MR 1688449, DOI https://doi.org/10.1007/s002220050311

S. M. Bhatwadekar and Raja Sridharan, The Euler class group of a Noetherian ring, Compositio Math. 122 (2000), no. 2, 183–222. MR 1775418, DOI https://doi.org/10.1023/A%3A1001872132498

Federico Binda and Amalendu Krishna, Zero cycles with modulus and zero cycles on singular varieties, Compos. Math. 154 (2018), no. 1, 120–187. MR 3719246, DOI https://doi.org/10.1112/S0010437X17007503

Federico Binda and Shuji Saito, Relative cycles with moduli and regulator maps, J. Inst. Math. Jussieu 18 (2019), no. 6, 1233–1293. MR 4021105, DOI https://doi.org/10.1017/s1474748017000391

J. G. Biswas and V. Srinivas, The Chow ring of a singular surface, Proc. Indian Acad. Sci. Math. Sci. 108 (1998), no. 3, 227–249. MR 1663739, DOI https://doi.org/10.1007/BF02844480

Spencer Bloch, $K{\rm _{2}}$ and algebraic cycles, Ann. of Math. (2) 99 (1974), 349–379. MR 342514, DOI https://doi.org/10.2307/1970902

Alberto Collino, Quillen’s ${\cal K}$-theory and algebraic cycles on almost nonsingular varieties, Illinois J. Math. 25 (1981), no. 4, 654–666. MR 630843

Mrinal Kanti Das and Satya Mandal, A Riemann-Roch theorem, J. Algebra 301 (2006), no. 1, 148–164. MR 2230324, DOI https://doi.org/10.1016/j.jalgebra.2005.10.007

Mrinal Kanti Das and Md. Ali Zinna, “Strong” Euler class of a stably free module of odd rank, J. Algebra 432 (2015), 185–204. MR 3334145, DOI https://doi.org/10.1016/j.jalgebra.2015.03.007

Hélène Esnault, V. Srinivas, and Eckart Viehweg, The universal regular quotient of the Chow group of points on projective varieties, Invent. Math. 135 (1999), no. 3, 595–664. MR 1669284, DOI https://doi.org/10.1007/s002220050296

William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323

Henri Gillet, $K$-theory and intersection theory, Handbook of $K$-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 235–293. MR 2181825, DOI https://doi.org/10.1007/3-540-27855-9_7

Alexander Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154 (French). MR 116023

Irving Kaplansky, Commutative rings, Allyn and Bacon, Inc., Boston, Mass., 1970. MR 0254021

Kazuya Kato, Milnor $K$-theory and the Chow group of zero cycles, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 241–253. MR 862638, DOI https://doi.org/10.1090/conm/055.1/862638

Kazuya Kato and Shuji Saito, Global class field theory of arithmetic schemes, Applications of algebraic $K$-theory to algebraic geometry and number theory, Part I, II (Boulder, Colo., 1983) Contemp. Math., vol. 55, Amer. Math. Soc., Providence, RI, 1986, pp. 255–331. MR 862639, DOI https://doi.org/10.1090/conm/055.1/862639

Moritz Kerz, The Gersten conjecture for Milnor $K$-theory, Invent. Math. 175 (2009), no. 1, 1–33. MR 2461425, DOI https://doi.org/10.1007/s00222-008-0144-8

Moritz Kerz, Milnor $K$-theory of local rings with finite residue fields, J. Algebraic Geom. 19 (2010), no. 1, 173–191. MR 2551760, DOI https://doi.org/10.1090/S1056-3911-09-00514-1

Moritz Kerz and Shuji Saito, Chow group of 0-cycles with modulus and higher-dimensional class field theory, Duke Math. J. 165 (2016), no. 15, 2811–2897. MR 3557274, DOI https://doi.org/10.1215/00127094-3644902

Moritz Kerz, Florian Strunk, and Georg Tamme, Algebraic $K$-theory and descent for blow-ups, Invent. Math. 211 (2018), no. 2, 523–577. MR 3748313, DOI https://doi.org/10.1007/s00222-017-0752-2

Amalendu Krishna, On 0-cycles with modulus, Algebra Number Theory 9 (2015), no. 10, 2397–2415. MR 3437766, DOI https://doi.org/10.2140/ant.2015.9.2397

Amalendu Krishna, Torsion in the 0-cycle group with modulus, Algebra Number Theory 12 (2018), no. 6, 1431–1469. MR 3864203, DOI https://doi.org/10.2140/ant.2018.12.1431

Amalendu Krishna, Murthy’s conjecture on 0-cycles, Invent. Math. 217 (2019), no. 2, 549–602. MR 3987177, DOI https://doi.org/10.1007/s00222-019-00871-8

Amalendu Krishna and Jinhyun Park, A module structure and a vanishing theorem for cycles with modulus, Math. Res. Lett. 24 (2017), no. 4, 1147–1176. MR 3723807, DOI https://doi.org/10.4310/MRL.2017.v24.n4.a10

Amalendu Krishna and V. Srinivas, Zero-cycles and $K$-theory on normal surfaces, Ann. of Math. (2) 156 (2002), no. 1, 155–195. MR 1935844, DOI https://doi.org/10.2307/3597187

Serge Lang, Abelian varieties, Interscience Tracts in Pure and Applied Mathematics, No. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1959. MR 0106225

M. Levine, A geometric theory of the Chow ring of a singular variety, unpublished manuscript.

Marc Levine, Bloch’s formula for singular surfaces, Topology 24 (1985), no. 2, 165–174. MR 793182, DOI https://doi.org/10.1016/0040-9383%2885%2990053-9

Marc Levine, Zero-cycles and $K$-theory on singular varieties, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 451–462. MR 927992

Marc Levine and Chuck Weibel, Zero cycles and complete intersections on singular varieties, J. Reine Angew. Math. 359 (1985), 106–120. MR 794801

Satya Mandal, Complete intersection $K$-theory and Chern classes, Math. Z. 227 (1998), no. 3, 423–454. MR 1612665, DOI https://doi.org/10.1007/PL00004384

Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273

John Milnor, Introduction to algebraic $K$-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72. MR 0349811

M. Pavaman Murthy, Zero cycles and projective modules, Ann. of Math. (2) 140 (1994), no. 2, 405–434. MR 1298718, DOI https://doi.org/10.2307/2118605

Claudio Pedrini and Charles Weibel, Divisibility in the Chow group of zero-cycles on a singular surface, Astérisque 226 (1994), 10–11, 371–409. $K$-theory (Strasbourg, 1992). MR 1317125

Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 85–147. Lecture Notes in Math., Vol. 341. MR 0338129

Markus Rost, Chow groups with coefficients, Doc. Math. 1 (1996), No. 16, 319–393. MR 1418952

Kay Rülling and Shuji Saito, Higher Chow groups with modulus and relative Milnor $K$-theory, Trans. Amer. Math. Soc. 370 (2018), no. 2, 987–1043. MR 3729494, DOI https://doi.org/10.1090/tran/7018

Marco Schlichting, Euler class groups and the homology of elementary and special linear groups, Adv. Math. 320 (2017), 1–81. MR 3709100, DOI https://doi.org/10.1016/j.aim.2017.08.034

V. Srinivas, Zero cycles on a singular surface. II, J. Reine Angew. Math. 362 (1985), 4–27. MR 809963, DOI https://doi.org/10.1515/crll.1985.362.4

Vasudevan Srinivas, Algebraic cycles on singular varieties, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 603–623. MR 2827811

A. A. Suslin, A cancellation theorem for projective modules over algebras, Dokl. Akad. Nauk SSSR 236 (1977), no. 4, 808–811 (Russian). MR 0466104

Richard G. Swan, A cancellation theorem for projective modules in the metastable range, Invent. Math. 27 (1974), 23–43. MR 376681, DOI https://doi.org/10.1007/BF01389963

R. W. Thomason and Thomas Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III, Progr. Math., vol. 88, Birkhäuser Boston, Boston, MA, 1990, pp. 247–435. MR 1106918, DOI https://doi.org/10.1007/978-0-8176-4576-2_10

Wilberd van der Kallen, A module structure on certain orbit sets of unimodular rows, J. Pure Appl. Algebra 57 (1989), no. 3, 281–316. MR 987316, DOI https://doi.org/10.1016/0022-4049%2889%2990035-2

Wilberd van der Kallen, Extrapolating an Euler class, J. Algebra 434 (2015), 65–71. MR 3342385, DOI https://doi.org/10.1016/j.jalgebra.2015.04.001

C. A. Weibel, Mayer-Vietoris sequences and module structures on $NK_ \ast $, Algebraic $K$-theory, Evanston 1980 (Proc. Conf., Northwestern Univ., Evanston, Ill., 1980) Lecture Notes in Math., vol. 854, Springer, Berlin, 1981, pp. 466–493. MR 618317, DOI https://doi.org/10.1007/BFb0089534

Additional Information

Rahul Gupta
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India
Email: Rahul.Gupta@mathematik.uni-regensburg.de

Amalendu Krishna
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Colaba, Mumbai, India
MR Author ID: 703987
Email: amal@math.tifr.res.in

Received by editor(s): May 10, 2018
Received by editor(s) in revised form: September 11, 2018
Published electronically: November 4, 2019
Article copyright: © Copyright 2019 University Press, Inc.