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.