Bloch’s formula for 0-cycles with modulus and higher-dimensional class field theory
Authors:
Federico Binda, Amalendu Krishna and Shuji Saito
Journal:
J. Algebraic Geom. 32 (2023), 323-384
DOI:
https://doi.org/10.1090/jag/792
Published electronically:
August 29, 2022
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We prove Bloch’s formula for the Chow group of 0-cycles with modulus on a smooth quasi-projective surface over a field. We use this formula to give a simple proof of the rank one case of a conjecture of Deligne and Drinfeld on lisse $\overline {\mathbb {Q}}_{\ell }$-sheaves. This was originally solved by Kerz and Saito in characteristic $\neq 2$.
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 10.1080/00927877908822375
- 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 10.1112/S0010437X17007503
- F. Binda and A. Krishna, Rigidity for relative $0$-cycles, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XXII (2021), 281–325; arXiv:1802.00165v3 [math.AG], (2020).
- Federico Binda and Amalendu Krishna, Zero-cycle groups on algebraic varieties, J. Éc. polytech. Math. 9 (2022), 281–325 (English, with English and French summaries). MR 4373665, DOI 10.5802/jep.183
- 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 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 10.1007/BF02844480
- Spencer Bloch, $K{\rm _{2}}$ and algebraic cycles, Ann. of Math. (2) 99 (1974), 349–379. MR 342514, DOI 10.2307/1970902
- Spencer Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986), no. 3, 267–304. MR 852815, DOI 10.1016/0001-8708(86)90081-2
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- 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
- Michael R. Stein and R. Keith Dennis, $K_{2}$ of radical ideals and semi-local rings revisited, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 281–303. MR 0406998
- Vladimir Drinfeld, On a conjecture of Deligne, Mosc. Math. J. 12 (2012), no. 3, 515–542, 668 (English, with English and Russian summaries). MR 3024821, DOI 10.17323/1609-4514-2012-12-3-515-542
- Hélène Esnault and Moritz Kerz, A finiteness theorem for Galois representations of function fields over finite fields (after Deligne), Acta Math. Vietnam. 37 (2012), no. 4, 531–562. MR 3058662
- 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 10.1007/s002220050296
- William Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147–167. MR 404257, DOI 10.1007/BF02684300
- 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, DOI 10.1007/978-1-4612-1700-8
- M. Ghosh and A. Krishna, Bertini theorems revisited, arXiv:1912.09076v2 [math.AG], 2020.
- Ulrich Görtz and Torsten Wedhorn, Algebraic geometry I, Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden, 2010. Schemes with examples and exercises. MR 2675155, DOI 10.1007/978-3-8348-9722-0
- Rahul Gupta and Amalendu Krishna, $K$-theory and 0-cycles on schemes, J. Algebraic Geom. 29 (2020), no. 3, 547–601. MR 4158460, DOI 10.1090/jag/744
- Rahul Gupta and Amalendu Krishna, Reciprocity for Kato-Saito idele class group with modulus, J. Algebra 608 (2022), 487–552. MR 4447740, DOI 10.1016/j.jalgebra.2022.06.004
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR 725671
- 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 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 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 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 10.1090/S1056-3911-09-00514-1
- Moritz Kerz and Shuji Saito, Lefschetz theorem for abelian fundamental group with modulus, Algebra Number Theory 8 (2014), no. 3, 689–701. MR 3218806, DOI 10.2140/ant.2014.8.689
- 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 10.1215/00127094-3644902
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
- Amalendu Krishna, On 0-cycles with modulus, Algebra Number Theory 9 (2015), no. 10, 2397–2415. MR 3437766, DOI 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 10.2140/ant.2018.12.1431
- 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 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 10.2307/3597187
- Marc Levine, Bloch’s formula for singular surfaces, Topology 24 (1985), no. 2, 165–174. MR 793182, DOI 10.1016/0040-9383(85)90053-9
- M. Levine, A geometric theory of the Chow ring of a singular variety, Unpublished manuscript, 1985.
- 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
- 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
- Claudio Pedrini and Charles A. Weibel, $K$-theory and Chow groups on singular varieties, 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. 339–370. MR 862641, DOI 10.1090/conm/055.1/862641
- Bjorn Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), no. 3, 1099–1127. MR 2144974, DOI 10.4007/annals.2004.160.1099
- Daniel Quillen, Higher algebraic $K$-theory. I, Algebraic $K$-theory, I: Higher $K$-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 341, Springer, Berlin, 1973, pp. 85–147. MR 0338129
- 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 10.1090/tran/7018
- Henrik Russell, Albanese varieties with modulus over a perfect field, Algebra Number Theory 7 (2013), no. 4, 853–892. MR 3095229, DOI 10.2140/ant.2013.7.853
- Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618
- The Stacks Project authors, Tag 004Z, Stacks Project, 2019.
- The Stacks Project authors, More on morphisms, Stacks Project, 2019.
- The Stacks Project authors, Commutative algebra, Stacks Project, 2019.
- Tamás Szamuely, Galois groups and fundamental groups, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, Cambridge, 2009. MR 2548205, DOI 10.1017/CBO9780511627064
- R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987), no. 1, 16–34. MR 893468, DOI 10.1016/0001-8708(87)90016-8
- 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 10.1007/978-0-8176-4576-2_{1}0
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 10.1080/00927877908822375
- 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 10.1112/S0010437X17007503
- F. Binda and A. Krishna, Rigidity for relative $0$-cycles, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XXII (2021), 281–325; arXiv:1802.00165v3 [math.AG], (2020).
- Federico Binda and Amalendu Krishna, Zero-cycle groups on algebraic varieties, J. Éc. Polytech. Math. 9 (2022), 281–325 (English, with English and French summaries). MR 4373665, DOI 10.5802/jep.183
- 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 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 10.1007/BF02844480
- Spencer Bloch, $K_{\mathrm {2}}$ and algebraic cycles, Ann. of Math. (2) 99 (1974), 349–379. MR 342514, DOI 10.2307/1970902
- Spencer Bloch, Algebraic cycles and higher $K$-theory, Adv. in Math. 61 (1986), no. 3, 267–304. MR 852815, DOI 10.1016/0001-8708(86)90081-2
- Armand Borel, Linear algebraic groups, 2nd ed., Graduate Texts in Mathematics, vol. 126, Springer-Verlag, New York, 1991. MR 1102012, DOI 10.1007/978-1-4612-0941-6
- Alberto Collino, Quillen’s ${\mathcal {K}}$-theory and algebraic cycles on almost nonsingular varieties, Illinois J. Math. 25 (1981), no. 4, 654–666. MR 630843
- Michael R. Stein and R. Keith Dennis, $K_{2}$ of radical ideals and semi-local rings revisited, Algebraic $K$-theory, II: “Classical” algebraic $K$-theory and connections with arithmetic (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972) Springer, Berlin, 1973, pp. 281–303. Lecture Notes in Math. Vol. 342. MR 0406998
- Vladimir Drinfeld, On a conjecture of Deligne, Mosc. Math. J. 12 (2012), no. 3, 515–542, 668 (English, with English and Russian summaries). MR 3024821, DOI 10.17323/1609-4514-2012-12-3-515-542
- Hélène Esnault and Moritz Kerz, A finiteness theorem for Galois representations of function fields over finite fields (after Deligne), Acta Math. Vietnam. 37 (2012), no. 4, 531–562. MR 3058662
- 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 10.1007/s002220050296
- William Fulton, Rational equivalence on singular varieties, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 147–167. MR 404257
- 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, DOI 10.1007/978-1-4612-1700-8
- M. Ghosh and A. Krishna, Bertini theorems revisited, arXiv:1912.09076v2 [math.AG], 2020.
- Ulrich Görtz and Torsten Wedhorn, Algebraic geometry I: Schemes with examples and exercises, Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden, 2010. MR 2675155, DOI 10.1007/978-3-8348-9722-0
- Rahul Gupta and Amalendu Krishna, $K$-theory and 0-cycles on schemes, J. Algebraic Geom. 29 (2020), no. 3, 547–601. MR 4158460, DOI 10.1090/jag/744
- Rahul Gupta and Amalendu Krishna, Reciprocity for Kato-Saito idele class group with modulus, J. Algebra 608 (2022), 487–552. MR 4447740, DOI 10.1016/j.jalgebra.2022.06.004
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- Jean-Pierre Jouanolou, Théorèmes de Bertini et applications, Progress in Mathematics, vol. 42, Birkhäuser Boston, Inc., Boston, MA, 1983 (French). MR 725671
- 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 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 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 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 10.1090/S1056-3911-09-00514-1
- Moritz Kerz and Shuji Saito, Lefschetz theorem for abelian fundamental group with modulus, Algebra Number Theory 8 (2014), no. 3, 689–701. MR 3218806, DOI 10.2140/ant.2014.8.689
- 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 10.1215/00127094-3644902
- Steven L. Kleiman, The transversality of a general translate, Compositio Math. 28 (1974), 287–297. MR 360616
- Amalendu Krishna, On $0$-cycles with modulus, Algebra Number Theory 9 (2015), no. 10, 2397–2415. MR 3437766, DOI 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 10.2140/ant.2018.12.1431
- 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 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 10.2307/3597187
- Marc Levine, Bloch’s formula for singular surfaces, Topology 24 (1985), no. 2, 165–174. MR 793182, DOI 10.1016/0040-9383(85)90053-9
- M. Levine, A geometric theory of the Chow ring of a singular variety, Unpublished manuscript, 1985.
- 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
- 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
- Claudio Pedrini and Charles A. Weibel, $K$-theory and Chow groups on singular varieties, 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. 339–370. MR 862641, DOI 10.1090/conm/055.1/862641
- Bjorn Poonen, Bertini theorems over finite fields, Ann. of Math. (2) 160 (2004), no. 3, 1099–1127. MR 2144974, DOI 10.4007/annals.2004.160.1099
- 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
- 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 10.1090/tran/7018
- Henrik Russell, Albanese varieties with modulus over a perfect field, Algebra Number Theory 7 (2013), no. 4, 853–892. MR 3095229, DOI 10.2140/ant.2013.7.853
- Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618
- The Stacks Project authors, Tag 004Z, Stacks Project, 2019.
- The Stacks Project authors, More on morphisms, Stacks Project, 2019.
- The Stacks Project authors, Commutative algebra, Stacks Project, 2019.
- Tamás Szamuely, Galois groups and fundamental groups, Cambridge Studies in Advanced Mathematics, vol. 117, Cambridge University Press, Cambridge, 2009. MR 2548205, DOI 10.1017/CBO9780511627064
- R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987), no. 1, 16–34. MR 893468, DOI 10.1016/0001-8708(87)90016-8
- 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 10.1007/978-0-8176-4576-2_10
Additional Information
Federico Binda
Affiliation:
Dipartimento di Matematica “Federigo Enriques”, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
MR Author ID:
1183287
ORCID:
0000-0002-3476-440X
Email:
federico.binda@unimi.it
Amalendu Krishna
Affiliation:
Department of Mathematics, Indian Institute of Science Bangalore, 560012, India
MR Author ID:
703987
Email:
amalenduk@iisc.ac.in
Shuji Saito
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo 153-8914, Japan
MR Author ID:
188665
ORCID:
0000-0002-6914-0242
Email:
sshuji@msb.biglobe.ne.jp
Received by editor(s):
January 27, 2021
Published electronically:
August 29, 2022
Additional Notes:
The first author was supported by the DFG SFB/CRC 1085 “Higher Invariants”. The third author was supported by JSPS KAKENHI Grant (15H03606) and the DFG SFB/CRC 1085 “Higher Invariants”
Article copyright:
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University Press, Inc.