Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



The two-dimensional Contou-Carrère symbol and reciprocity laws

Authors: Denis Osipov and Xinwen Zhu
Journal: J. Algebraic Geom. 25 (2016), 703-774
Published electronically: April 22, 2016
MathSciNet review: 3533184
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Abstract | References | Additional Information

Abstract: We define a two-dimensional Contou-Carrère symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrère symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We also relate the two-dimensional Contou-Carrère symbol to the two-dimensional class field theory.

References [Enhancements On Off] (What's this?)

  • [AP] Greg W. Anderson and Fernando Pablos Romo, Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an Artinian local ring, Comm. Algebra 32 (2004), no. 1, 79-102. MR 2036223 (2005d:11099),
  • [BBE] Alexander Beilinson, Spencer Bloch, and Hélène Esnault, $ \epsilon $-factors for Gauss-Manin determinants, Dedicated to Yuri I. Manin on the occasion of his 65th birthday, Mosc. Math. J. 2 (2002), no. 3, 477-532. MR 1988970 (2004m:14011)
  • [Br1] Lawrence Breen, Bitorseurs et cohomologie non abélienne, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 401-476 (French). MR 1086889 (92m:18019)
  • [Br2] Lawrence Breen, On the classification of $ 2$-gerbes and $ 2$-stacks, Astérisque 225 (1994), 160 (English, with English and French summaries). MR 1301844 (95m:18006)
  • [BM] J.-L. Brylinski and D. A. McLaughlin, The geometry of two-dimensional symbols, $ K$-Theory 10 (1996), no. 3, 215-237. MR 1394378 (97e:19003),
  • [CC1] Carlos Contou-Carrère, Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole modéré, C. R. Acad. Sci. Paris Sér. I Math. 318 (1994), no. 8, 743-746 (French, with English and French summaries). MR 1272340 (95c:14059)
  • [CC2] Carlos Contou-Carrère, Jacobienne locale d'une courbe formelle relative, Rend. Semin. Mat. Univ. Padova 130 (2013), 1-106 (French, with English summary). MR 3148632,
  • [Del] P. Deligne, Le symbole modéré, Inst. Hautes Études Sci. Publ. Math. 73 (1991), 147-181 (French). MR 1114212 (93i:14030)
  • [Dr] Vladimir Drinfeld, Infinite-dimensional vector bundles in algebraic geometry: an introduction, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 263-304. MR 2181808 (2007d:14038),
  • [FZ] Edward Frenkel and Xinwen Zhu, Gerbal representations of double loop groups, Int. Math. Res. Not. IMRN 17 (2012), 3929-4013. MR 2972546,
  • [G] D. Gaitsgory, Affine Grassmannian and the loop group, Seminar Notes written by D. Gaitsgory and N. Rozenblyum, 2009, 12 pp., available at http://$ \sim $gaitsgde/grad_2009/SeminarNotes/Oct13(AffGr).pdf
  • [Gr] Daniel Grayson, Higher algebraic $ K$-theory. II (after Daniel Quillen), Algebraic $ K$-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), Lecture Notes in Math., Vol. 551, Springer, Berlin, 1976, pp. 217-240. MR 0574096 (58 #28137)
  • [HL] I. Horozov and Zh. Luo, On the Contou-Carrere symbol for surfaces, preprint, arXiv:1310.7065 [math.AG].
  • [Kap] M. Kapranov, Semiinfinite symmetric powers, preprint, arXiv:math/0107089 [math.QA].
  • [KV] Mikhail Kapranov and Éric Vasserot, Formal loops. II. A local Riemann-Roch theorem for determinantal gerbes, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 1, 113-133 (English, with English and French summaries). MR 2332353 (2008k:14048),
  • [Ka1] 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 (83g:12020a)
  • [Ka2] Kazuya Kato, Generalized class field theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 419-428. MR 1159230 (93e:19011)
  • [KR] Kazuya Kato and Henrik Russell, Modulus of a rational map into a commutative algebraic group, Kyoto J. Math. 50 (2010), no. 3, 607-622. MR 2723864 (2011j:14098),
  • [KS] Kazuya Kato and Shuji Saito, Two-dimensional class field theory, Galois groups and their representations (Nagoya, 1981) Adv. Stud. Pure Math., vol. 2, North-Holland, Amsterdam, 1983, pp. 103-152. MR 732466 (87a:11060)
  • [Mor] Jack Morava, An algebraic analog of the Virasoro group, Quantum groups and integrable systems (Prague, 2001), Czechoslovak J. Phys. 51 (2001), no. 12, 1395-1400. MR 1917711 (2003g:22026),
  • [O1] D. V. Osipov, Adelic constructions of direct images of differentials and symbols, Mat. Sb. 188 (1997), no. 5, 59-84 (Russian, with Russian summary); English transl., Sb. Math. 188 (1997), no. 5, 697-723. MR 1478630 (98k:11093),
  • [O2] D. V. Osipov, Central extensions and reciprocity laws on algebraic surfaces, Mat. Sb. 196 (2005), no. 10, 111-136 (Russian, with Russian summary); English transl., Sb. Math. 196 (2005), no. 9-10, 1503-1527. MR 2195664 (2007f:11071),
  • [O3] Denis Osipov, Adeles on $ n$-dimensional schemes and categories $ C_n$, Internat. J. Math. 18 (2007), no. 3, 269-279. MR 2314612 (2008b:14005),
  • [O4] Denis V. Osipov, $ n$-dimensional local fields and adeles on $ n$-dimensional schemes, Surveys in contemporary mathematics, London Math. Soc. Lecture Note Ser., vol. 347, Cambridge Univ. Press, Cambridge, 2008, pp. 131-164. MR 2388492 (2009b:14044)
  • [OsZh] 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 (2012h:19010),
  • [Pal] Ambrus Pál, On the kernel and the image of the rigid analytic regulator in positive characteristic, Publ. Res. Inst. Math. Sci. 46 (2010), no. 2, 255-288. MR 2722779 (2011g:19006),
  • [Pa1] A. N. Paršin, Class fields and algebraic $ K$-theory, Uspehi Mat. Nauk 30 (1975), no. 1 (181), 253-254 (Russian). MR 0401710 (53 #5537)
  • [Pa2] A. N. Paršin, Abelian coverings of arithmetic schemes, Dokl. Akad. Nauk SSSR 243 (1978), no. 4, 855-858 (Russian). MR 514485 (80b:14014)
  • [Pa3] A. N. Parshin, Local class field theory, Algebraic geometry and its applications, Trudy Mat. Inst. Steklov. 165 (1984), 143-170 (Russian). MR 752939 (85m:11086)
  • [S] V. Srinivas, Algebraic $ K$-theory, 2nd ed., Progress in Mathematics, vol. 90, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1382659 (97c:19001)

Additional Information

Denis Osipov
Affiliation: Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia – and – National University of Science and Technology MISIS, Leninskii pr. 4, Moscow, 119049 Russia

Xinwen Zhu
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208
Address at time of publication: Department of Mathematics, California Institute of Technology, 1200 E. California Boulevard, Pasadena, California 91125

Received by editor(s): July 28, 2013
Received by editor(s) in revised form: February 14, 2014, and May 6, 2014
Published electronically: April 22, 2016
Additional Notes: The first author was supported by the Russian Foundation for Basic Research (grants No. 14-01-00178-a, No. 13-01-12420 ofi_m2, and No. 12-01-33024 mol_a_ved) and by the Programme for the Support of Leading Scientific Schools of the Russian Federation (grant No. NSh-2998.2014.1). The second author was partially supported by NSF grants DMS-1001280/1313894 and DMS-1303296 and by an AMS Centennial Fellowship
Article copyright: © Copyright 2016 University Press, Inc.

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