Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T04:02:43.691Z Has data issue: false hasContentIssue false
  • English
  • Français

The strong Suslin reciprocity law

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 April 2021

Daniil Rudenko
Daniil Rudenko*
Affiliation:
Department of Mathematics, The University of Chicago, Eckhart Hall, 5734 S University Ave, Chicago, IL60637, [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the strong Suslin reciprocity law conjectured by A. Goncharov. The Suslin reciprocity law is a generalization of the Weil reciprocity law to higher Milnor $K$-theory. The Milnor $K$-groups can be identified with the top cohomology groups of the polylogarithmic motivic complexes; Goncharov's conjecture predicts the existence of a contracting homotopy underlying Suslin reciprocity. The main ingredient of the proof is a homotopy invariance theorem for the cohomology of the polylogarithmic motivic complexes in the ‘next to Milnor’ degree. We apply these results to the theory of scissors congruences of hyperbolic polytopes. For every triple of rational functions on a compact projective curve over $\mathbb {C}$ we construct a hyperbolic polytope (defined up to scissors congruence). The hyperbolic volume and the Dehn invariant of this polytope can be computed directly from the triple of rational functions on the curve.

Keywords

polylogarithmsK-theoryscissors congruence

MSC classification

Primary: 11G55: Polylogarithms and relations with $K$-theory
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 4 , April 2021 , pp. 649 - 676
Check for updates
Copyright
© The Author(s) 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bass, H. and Tate, J., The Milnor ring of a global field, in ‘Classical’ algebraic K-theory, and connections with arithmetic, Lecture Notes in Mathematics, vol. 342 (Springer, Berlin, 1973), 349–446.Google Scholar
Beilinson, A. A., Height pairing on algebraic cycles, in Current trends in arithmetical algebraic geometry, Contemporary Mathematics, vol. 67 (American Mathematical Society, Providence, RI, 1987), 1–24.Google Scholar
Deligne, P., Théorie de Hodge: II, Publ. Math. Inst. Hautes Études Sci., 40 (1971), 5–57.CrossRefGoogle Scholar
Dupont, J., Scissors congruences, group homology and characteristic classes (World Scientific, River Edge, NJ, 2001).CrossRefGoogle Scholar
Goncharov, A. B., Polylogarithms and motivic Galois groups, Proc. Sympos. Pure Math. 55 (1994), 43–96.CrossRefGoogle Scholar
Goncharov, A. B., Polylogarithms in arithmetic and geometry, in Proceedings of the International Congress of Mathematicians, ed. S. D. Chatterji (Birkhäuser, Basel, 1995).CrossRefGoogle Scholar
Goncharov, A. B., Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995), 197–318.CrossRefGoogle Scholar
Goncharov, A. B., Polylogarithms, regulators and Arakelov motivic complexes, J. Amer. Math. Soc. 18 (2005), 1–60.CrossRefGoogle Scholar
Milnor, J., Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1970), 318–344.CrossRefGoogle Scholar
Rost, M., Chow groups with coefficients, Doc. Math. 1 (1996), 319–393.Google Scholar
Suslin, A. A., Reciprocity laws and the stable rank of polynomial rings, Izv. Akad. Nauk Ser. Mat. 43 (1979), 1394–1429.Google Scholar
Suslin, A. A., Homology of $GL_n$, characteristic classes and Milnor $K$-theory, Proc. Steklov Inst. Math. 165 (1985), 207–226.Google Scholar
Suslin, A. A., $K_3$ of a field, and the Bloch group, Proc. Steklov Inst. Math. 183 (1991), 217–239.Google Scholar
Sydler, J.-P., Conditions nécessaires et suffisantes pour l'equivalence des polyèdres de l'espace euclidien à trois dimensions, Comment. Math. Helv. 40 (1965), 43–80.CrossRefGoogle Scholar