A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions

Braunling, Oliver; Groechenig, Michael; Wolfson, Jesse

doi:10.1090/btran/81

A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions
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Trans. Amer. Math. Soc. Ser. B 8 (2021), 679-753

Abstract:

We generalize Contou-Carrère symbols to higher dimensions. To an $(n+1)$-tuple $f_0,\dots ,f_n \in A((t_1))\cdots ((t_n))^{\times }$, where $A$ denotes an algebra over a field $k$, we associate an element $(f_0,\dots ,f_n) \in A^{\times }$, extending the higher tame symbol for $k = A$, and earlier constructions for $n = 1$ by Contou-Carrère, and $n = 2$ by Osipov–Zhu. It is based on the concept of higher commutators for central extensions by spectra. Using these tools, we describe the higher Contou-Carrère symbol as a composition of boundary maps in algebraic $K$-theory, and prove a version of Parshin–Kato reciprocity for higher Contou-Carrère symbols.

References

John Frank Adams, Infinite loop spaces, Annals of Mathematics Studies, No. 90, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1978. MR 505692, DOI 10.1515/9781400821259
E. Arbarello, C. De Concini, and V. G. Kac, The infinite wedge representation and the reciprocity law for algebraic curves, Theta functions—Bowdoin 1987, Part 1 (Brunswick, ME, 1987) Proc. Sympos. Pure Math., vol. 49, Amer. Math. Soc., Providence, RI, 1989, pp. 171–190. MR 1013132, DOI 10.1007/bf01228409
M. Artin, A. Grothendieck, and J.-L. Verdier, Theorie de topos et cohomologie etale des schemas I, II, III, Lecture Notes in Mathematics, vol. 269, 270, 305, Springer, 1971.
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, DOI 10.1081/AGB-120027853
Leovigildo Alonso Tarrío, Ana Jeremías López, and Joseph Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 1, 1–39. MR 1422312, DOI 10.1016/S0012-9593(97)89914-4
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
Alexander Beilinson, Spencer Bloch, and Hélène Esnault, $\epsilon$-factors for Gauss-Manin determinants, Mosc. Math. J. 2 (2002), no. 3, 477–532. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1988970, DOI 10.17323/1609-4514-2002-2-3-477-532
Spencer Bloch and Hélène Esnault, Gauss-Manin determinants for rank 1 irregular connections on curves, Math. Ann. 321 (2001), no. 1, 15–87. With an appendix in French by P. Deligne. MR 1857369, DOI 10.1007/PL00004499
A. A. Beĭlinson, Residues and adèles, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 44–45 (Russian). MR 565095
A. A. Beĭlinson, How to glue perverse sheaves, $K$-theory, arithmetic and geometry (Moscow, 1984–1986) Lecture Notes in Math., vol. 1289, Springer, Berlin, 1987, pp. 42–51. MR 923134, DOI 10.1007/BFb0078366
Andrew J. Blumberg, David Gepner, and Gonçalo Tabuada, A universal characterization of higher algebraic $K$-theory, Geom. Topol. 17 (2013), no. 2, 733–838. MR 3070515, DOI 10.2140/gt.2013.17.733
O. Braunling, M. Groechenig, and J. Wolfson, Geometric and analytic structures on the higher adèles, Res. Math. Sci. 3 (2016), Paper No. 22, 56. MR 3536437, DOI 10.1186/s40687-016-0064-y
O. Braunling, M. Groechenig, and J. Wolfson, Operator ideals in Tate objects, Math. Res. Lett. 23 (2016), no. 6, 1565–1631. MR 3621099, DOI 10.4310/MRL.2016.v23.n6.a2
Oliver Braunling, Michael Groechenig, and Jesse Wolfson, Tate objects in exact categories, Mosc. Math. J. 16 (2016), no. 3, 433–504. With an appendix by Jan Šťovíček and Jan Trlifaj. MR 3510209, DOI 10.17323/1609-4514-2016-16-3-433-504
Oliver Braunling, Michael Groechenig, and Jesse Wolfson, The $A_\infty$-structure of the index map, Ann. K-Theory 3 (2018), no. 4, 581–614. MR 3892960, DOI 10.2140/akt.2018.3.581
Oliver Braunling, Michael Groechenig, and Jesse Wolfson, The index map in algebraic $K$-theory, Selecta Math. (N.S.) 24 (2018), no. 2, 1039–1091. MR 3782417, DOI 10.1007/s00029-017-0364-0
Oliver Braunling, On the local residue symbol in the style of Tate and Beilinson, New York J. Math. 24 (2018), 458–513. MR 3855636
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Carlos E. Contou-Carrère, La jacobienne généralisée d’une courbe relative; construction et propriété universelle de factorisation, C. R. Acad. Sci. Paris Sér. A-B 289 (1979), no. 3, A203–A206 (French, with English summary). MR 552212
Carlos E. Contou-Carrère, Corps de classes local géométrique relatif, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 9, 481–484 (French, with English summary). MR 612541
C. Contou-Carrère, Jacobiennes généralisées globales relatives, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 69–109 (French). MR 1106897
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
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, DOI 10.4171/RSMUP/130-1
Brian Conrad, Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc. 22 (2007), no. 3, 205–257. MR 2356346
P. Deligne, Le symbole modéré, Inst. Hautes Études Sci. Publ. Math. 73 (1991), 147–181 (French). MR 1114212, DOI 10.1007/BF02699258
W. G. Dwyer and J. P. C. Greenlees, Complete modules and torsion modules, Amer. J. Math. 124 (2002), no. 1, 199–220. MR 1879003, DOI 10.1353/ajm.2002.0001
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, DOI 10.1007/0-8176-4467-9_{7}
A. I. Efimov, Formal completion of a category along a subcategory, arXiv:1006.4721, 06 2010.
Thomas Geisser, Motivic cohomology, $K$-theory and topological cyclic homology, Handbook of $K$-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 193–234. MR 2181824, DOI 10.1007/3-540-27855-9_{6}
Henri Antoine Gillet, THE APPLICATIONS OF ALGEBRAIC K-THEORY TO INTERSECTION THEORY, ProQuest LLC, Ann Arbor, MI, 1978. Thesis (Ph.D.)–Harvard University. MR 2940798
Paul G. Goerss and John F. Jardine, Simplicial homotopy theory, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2009. Reprint of the 1999 edition [MR1711612]. MR 2840650, DOI 10.1007/978-3-0346-0189-4
J. P. C. Greenlees and J. P. May, Derived functors of $I$-adic completion and local homology, J. Algebra 149 (1992), no. 2, 438–453. MR 1172439, DOI 10.1016/0021-8693(92)90026-I
S. O. Gorchinskiĭ and D. V. Osipov, Explicit formula for the higher-dimensional Contou-Carrère symbol, Uspekhi Mat. Nauk 70 (2015), no. 1(421), 183–184 (Russian); English transl., Russian Math. Surveys 70 (2015), no. 1, 171–173. MR 3353121, DOI 10.4213/rm9653
S. O. Gorchinskiĭ and D. V. Osipov, A higher-dimensional Contou-Carrère symbol: local theory, Mat. Sb. 206 (2015), no. 9, 21–98 (Russian, with Russian summary); English transl., Sb. Math. 206 (2015), no. 9-10, 1191–1259. MR 3438594, DOI 10.4213/sm8516
Dennis Gaitsgory and Nick Rozenblyum, DG indschemes, Perspectives in representation theory, Contemp. Math., vol. 610, Amer. Math. Soc., Providence, RI, 2014, pp. 139–251. MR 3220628, DOI 10.1090/conm/610/12080
A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167. MR 217085
Pierre Colmez and Jean-Pierre Serre (eds.), Correspondance Grothendieck-Serre, Documents Mathématiques (Paris) [Mathematical Documents (Paris)], vol. 2, Société Mathématique de France, Paris, 2001 (French). MR 1942134
M. Groth, A short course on infinity-categories, arXiv:1007.2925, 07 2010.
Mark Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001), no. 1, 63–127. MR 1860878, DOI 10.1016/S0022-4049(00)00172-9
Mark Hovey, Brooke Shipley, and Jeff Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208. MR 1695653, DOI 10.1090/S0894-0347-99-00320-3
A. Huber, On the Parshin-Beĭlinson adèles for schemes, Abh. Math. Sem. Univ. Hamburg 61 (1991), 249–273. MR 1138291, DOI 10.1007/BF02950770
Florian Ivorra and Kay Rülling, K-groups of reciprocity functors, J. Algebraic Geom. 26 (2017), no. 2, 199–278. MR 3606996, DOI 10.1090/jag/678
M. M. Kapranov, Semi-infinite symmetric powers, arXiv:math/0107089v1 [math.QA], 2001.
Kazuya Kato, A generalization of local class field theory by using $K$-groups. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 26 (1979), no. 2, 303–376. MR 550688
Kazuya Kato, Class field theory and algebraic $K$-theory, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 109–126. MR 726423, DOI 10.1007/BFb0099960
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, Existence theorem for higher local fields, Invitation to higher local fields (Münster, 1999) Geom. Topol. Monogr., vol. 3, Geom. Topol. Publ., Coventry, 2000, pp. 165–195. MR 1804933, DOI 10.2140/gtm.2000.3.165
Bernhard Keller, On the cyclic homology of exact categories, J. Pure Appl. Algebra 136 (1999), no. 1, 1–56. MR 1667558, DOI 10.1016/S0022-4049(97)00152-7
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
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
Bruno Kahn, Shuji Saito, and Takao Yamazaki, Reciprocity sheaves, Compos. Math. 152 (2016), no. 9, 1851–1898. With two appendices by Kay Rülling. MR 3568941, DOI 10.1112/S0010437X16007466
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, DOI 10.1016/j.ansens.2006.12.003
Klaus Lamotke, Semisimpliziale algebraische Topologie, Die Grundlehren der mathematischen Wissenschaften, Band 147, Springer-Verlag, Berlin-New York, 1968 (German). MR 0245005, DOI 10.1007/978-3-662-12988-3
Jean-Louis Loday, $K$-théorie algébrique et représentations de groupes, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 309–377 (French). MR 447373, DOI 10.24033/asens.1312
J. Lurie, Higher algebra.
Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659, DOI 10.1515/9781400830558
J. Lurie, Derived algebraic geometry XII: Proper morphisms, completions, and the Grothendieck existence theorem, arXiv:0911.0018, 11 2009.
J. Lurie, (infinity,2)-categories and the Goodwillie Calculus I.
J. Peter May, $E_{\infty }$ ring spaces and $E_{\infty }$ ring spectra, Lecture Notes in Mathematics, Vol. 577, Springer-Verlag, Berlin-New York, 1977. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave. MR 0494077
J. Peter May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992. Reprint of the 1967 original. MR 1206474
John Milnor, Algebraic $K$-theory and quadratic forms, Invent. Math. 9 (1969/70), 318–344. MR 260844, DOI 10.1007/BF01425486
M. Morrow, An introduction to higher dimensional local fields and adèles, arXiv:1204.0586.
Carlo Mazza, Vladimir Voevodsky, and Charles Weibel, Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2006. MR 2242284
Evgeny Musicantov and Alexander Yom Din, Reciprocity laws and $K$-theory, Ann. K-Theory 2 (2017), no. 1, 27–46. MR 3599515, DOI 10.2140/akt.2017.2.27
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, DOI 10.1070/SM2005v196n10ABEH003710
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
Denis Osipov and Xinwen Zhu, The two-dimensional Contou-Carrère symbol and reciprocity laws, J. Algebraic Geom. 25 (2016), no. 4, 703–774. MR 3533184, DOI 10.1090/jag/664
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, DOI 10.2977/PRIMS/9
A. N. Paršin, Abelian coverings of arithmetic schemes, Dokl. Akad. Nauk SSSR 243 (1978), no. 4, 855–858 (Russian). MR 514485
A. N. Parshin, Local class field theory, Trudy Mat. Inst. Steklov. 165 (1984), 143–170 (Russian). Algebraic geometry and its applications. MR 752939
A. Parshin and T. Fimmel, Introduction to higher adelic theory (draft), unpublished, 1999.
Fernando Pablos Romo, On the tame symbol of an algebraic curve, Comm. Algebra 30 (2002), no. 9, 4349–4368. MR 1936475, DOI 10.1081/AGB-120013323
Luigi Previdi, Locally compact objects in exact categories, Internat. J. Math. 22 (2011), no. 12, 1787–1821. MR 2872533, DOI 10.1142/S0129167X11007379
Marco Porta, Liran Shaul, and Amnon Yekutieli, Completion by derived double centralizer, Algebr. Represent. Theory 17 (2014), no. 2, 481–494. MR 3181733, DOI 10.1007/s10468-013-9405-3
Marco Porta, Liran Shaul, and Amnon Yekutieli, Erratum to: On the homology of completion and torsion, Algebras and Representation Theory 18 (2015), no. 5, 1401–1405.
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
Pavel Safronov, Quasi-Hamiltonian reduction via classical Chern-Simons theory, Adv. Math. 287 (2016), 733–773. MR 3422691, DOI 10.1016/j.aim.2015.09.031
Sho Saito, On Previdi’s delooping conjecture for $K$-theory, Algebra Number Theory 9 (2015), no. 1, 1–11. MR 3317759, DOI 10.2140/ant.2015.9.1
Peter Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), no. 2, 161–180. MR 1973941, DOI 10.7146/math.scand.a-14399
Marco Schlichting, Delooping the $K$-theory of exact categories, Topology 43 (2004), no. 5, 1089–1103. MR 2079996, DOI 10.1016/j.top.2004.01.005
Marco Schlichting, Negative $K$-theory of derived categories, Math. Z. 253 (2006), no. 1, 97–134. MR 2206639, DOI 10.1007/s00209-005-0889-3
Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
John Tate, Residues of differentials on curves, Ann. Sci. École Norm. Sup. (4) 1 (1968), 149–159. MR 227171, DOI 10.24033/asens.1162
The Stacks Project Authors, Stacks project, http://math.columbia.edu/algebraic_geometry/stacks-git.
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
Friedhelm Waldhausen, Algebraic $K$-theory of spaces, Algebraic and geometric topology (New Brunswick, N.J., 1983) Lecture Notes in Math., vol. 1126, Springer, Berlin, 1985, pp. 318–419. MR 802796, DOI 10.1007/BFb0074449
Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324, DOI 10.1017/CBO9781139644136
Amnon Yekutieli, An explicit construction of the Grothendieck residue complex, Astérisque 208 (1992), 127 (English, with French summary). With an appendix by Pramathanath Sastry. MR 1213064