Chow dilogarithm and strong Suslin reciprocity law
Author:
Vasily Bolbachan
Journal:
J. Algebraic Geom. 32 (2023), 697-728
DOI:
https://doi.org/10.1090/jag/811
Published electronically:
May 18, 2023
Full-text PDF
Abstract |
References |
Additional Information
Abstract: We prove a conjecture of A. Goncharov concerning strong Suslin reciprocity law. The main idea of the proof is the construction of the norm map on so-called lifted reciprocity maps. This construction is similar to the construction of the norm map on Milnor $K$-theory. As an application, we express Chow dilogarithm in terms of Bloch-Wigner dilogarithm. Also, we obtain a new reciprocity law for four rational functions on an arbitrary algebraic surface with values in the pre-Bloch group.
References
- 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., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 349–446. MR 0442061, DOI 10.1007/BFb0073733
- Johan L. Dupont and Ebbe Thue Poulsen, Generation of $\textbf {C}(x)$ by a restricted set of operators, J. Pure Appl. Algebra 25 (1982), no. 2, 155–157. MR 662759, DOI 10.1016/0022-4049(82)90034-2
- A. B. Goncharov, Polylogarithms and motivic Galois groups, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43–96. MR 1265551, DOI 10.1090/pspum/055.2/1265551
- A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), no. 2, 197–318. MR 1348706, DOI 10.1006/aima.1995.1045
- A. B. Goncharov, Polylogarithms, regulators, and Arakelov motivic complexes, J. Amer. Math. Soc. 18 (2005), no. 1, 1–60. MR 2114816, DOI 10.1090/S0894-0347-04-00472-2
- Ivan Horozov, Reciprocity laws on algebraic surfaces via iterated integrals, J. K-Theory 14 (2014), no. 2, 273–312. With an appendix by Horozov and Matt Kerr. MR 3264264, DOI 10.1017/is014006014jkt271
- Kazuya Kato, A generalization of local class field theory by using $K$-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 3, 603–683. MR 603953
- János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- Matt Kerr, James Lewis, and Patrick Lopatto, Simplicial Abel-Jacobi maps and reciprocity laws, J. Algebraic Geom. 27 (2018), no. 1, 121–172. With an appendix by José Ignacio Burgos-Gil. MR 3722692, DOI 10.1090/jag/692
- John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
- Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
- Denis Osipov and Xinwen Zhu, A categorical proof of the Parshin reciprocity laws on algebraic surfaces, Algebra Number Theory 5 (2011), no. 3, 289–337. MR 2833793, DOI 10.2140/ant.2011.5.289
- A. N. Paršin, Class fields and algebraic $K$-theory, Uspehi Mat. Nauk 30 (1975), no. 1 (181), 253–254 (Russian). MR 0401710
- Daniil Rudenko, The strong Suslin reciprocity law, Compos. Math. 157 (2021), no. 4, 649–676. MR 4241111, DOI 10.1112/s0010437x20007666
- A. A. Suslin, Reciprocity laws and the stable rank of rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1394–1429 (Russian). MR 567040
- A. A. Suslin, $K_3$ of a field and the Bloch group, Proc. Steklov Inst. Math. 183 (1991), 217–239.
- Charles A. Weibel, The $K$-book, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. An introduction to algebraic $K$-theory. MR 3076731, DOI 10.1090/gsm/145
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
References
- 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., Battelle Memorial Inst., Seattle, Wash., 1972) Lecture Notes in Math., Vol. 342, Springer, Berlin, 1973, pp. 349–446. MR 0442061, DOI 10.1007/BFb0073733
- Johan L. Dupont and Ebbe Thue Poulsen, Generation of $\mathbf {C}(x)$ by a restricted set of operators, J. Pure Appl. Algebra 25 (1982), no. 2, 155–157. MR 662759, DOI 10.1016/0022-4049(82)90034-2
- A. B. Goncharov, Polylogarithms and motivic Galois groups, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 43–96. MR 1265551
- A. B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), no. 2, 197–318. MR 1348706, DOI 10.1006/aima.1995.1045
- A. B. Goncharov, Polylogarithms, regulators, and Arakelov motivic complexes, J. Amer. Math. Soc. 18 (2005), no. 1, 1–60. MR 2114816, DOI 10.1090/S0894-0347-04-00472-2
- Ivan Horozov, Reciprocity laws on algebraic surfaces via iterated integrals, J. K-Theory 14 (2014), no. 2, 273–312. With an appendix by Horozov and Matt Kerr. MR 3264264, DOI 10.1017/is014006014jkt271
- Kazuya Kato, A generalization of local class field theory by using $K$-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 3, 603–683. MR 603953
- János Kollár, Lectures on resolution of singularities, Annals of Mathematics Studies, vol. 166, Princeton University Press, Princeton, NJ, 2007. MR 2289519
- Matt Kerr, James Lewis, and Patrick Lopatto, Simplicial Abel-Jacobi maps and reciprocity laws, J. Algebraic Geom. 27 (2018), no. 1, 121–172. With an appendix by José Ignacio Burgos-Gil. MR 3722692, DOI 10.1090/jag/692
- John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
- Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
- Denis Osipov and Xinwen Zhu, A categorical proof of the Parshin reciprocity laws on algebraic surfaces, Algebra Number Theory 5 (2011), no. 3, 289–337. MR 2833793, DOI 10.2140/ant.2011.5.289
- A. N. Parshin, Class fields and algebraic $K$-theory, Uspehi Mat. Nauk 30 (1975), no. 1 (181), 253–254 (Russian). MR 0401710
- Daniil Rudenko, The strong Suslin reciprocity law, Compos. Math. 157 (2021), no. 4, 649–676. MR 4241111, DOI 10.1112/s0010437x20007666
- A. A. Suslin, Reciprocity laws and the stable rank of rings of polynomials, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 6, 1394–1429 (Russian). MR 567040
- A. A. Suslin, $K_3$ of a field and the Bloch group, Proc. Steklov Inst. Math. 183 (1991), 217–239.
- Charles A. Weibel, The $K$-book: An introduction to algebraic $K$-theory, Graduate Studies in Mathematics, vol. 145, American Mathematical Society, Providence, RI, 2013. MR 3076731, DOI 10.1090/gsm/145
- Oscar Zariski and Pierre Samuel, Commutative algebra, Volume I, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, New Jersey, 1958. With the cooperation of I. S. Cohen. MR 0090581
Additional Information
Vasily Bolbachan
Affiliation:
Skolkovo Institute of Science and Technology, Moscow, Russia; Faculty of Mathematics, National Research University Higher School of Ecnomics, Russian Federation, Usacheva str., 6, Moscow 119048, Russia; and HSE-Skoltech International Laboratory of Representation Theory and Mathematical Physics, Usacheva str., 6, Moscow 119048, Russia
MR Author ID:
1468287
ORCID:
0000-0001-6471-8669
Email:
vbolbachan@gmail.com
Received by editor(s):
May 8, 2021
Received by editor(s) in revised form:
October 28, 2021
Published electronically:
May 18, 2023
Additional Notes:
This paper was partially supported by the Basic Research Program at the HSE University and by the Moebius Contest Foundation for Young Scientists
Article copyright:
© Copyright 2023
University Press, Inc.
