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A generalized Contou-Carrère symbol and its reciprocity laws in higher dimensions. (English) Zbl 1505.19002

Summary: 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.

MSC:

19D45 Higher symbols, Milnor \(K\)-theory
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[1] Adams, John Frank, Infinite loop spaces, Annals of Mathematics Studies, No. 90, x+214 pp. (1978), Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo · Zbl 0398.55008
[2] 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. 1013132 (90i:22034) · Zbl 0699.22028
[3] 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.
[4] G. W. Anderson and F. 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. 2036223 (2005d:11099) · Zbl 1077.14033
[5] Alonso Tarr\'{\i }o, Leovigildo; Jerem\'{\i }as L\'{o}pez, Ana; Lipman, Joseph, Local homology and cohomology on schemes, Ann. Sci. \'{E}cole Norm. Sup. (4), 30, 1, 1-39 (1997) · Zbl 0894.14002
[6] Bass, Hyman, Algebraic \(K\)-theory, xx+762 pp. (1968), W. A. Benjamin, Inc., New York-Amsterdam · Zbl 0174.30302
[7] Beilinson, Alexander; Bloch, Spencer; Esnault, H\'{e}l\`ene, \( \epsilon \)-factors for Gauss-Manin determinants, Mosc. Math. J., 2, 3, 477-532 (2002) · Zbl 1061.14010
[8] Bloch, Spencer; Esnault, H\'{e}l\`ene, Gauss-Manin determinants for rank 1 irregular connections on curves, Math. Ann., 321, 1, 15-87 (2001) · Zbl 1040.14006
[9] Be\u{\i }linson, A. A., Residues and ad\`eles, Funktsional. Anal. i Prilozhen., 14, 1, 44-45 (1980) · Zbl 0509.14018
[10] Be\u{\i }linson, A. A., How to glue perverse sheaves. \(K\)-theory, arithmetic and geometry, Moscow, 1984-1986, Lecture Notes in Math. 1289, 42-51 (1987), Springer, Berlin · Zbl 0651.14009
[11] Blumberg, Andrew J.; Gepner, David; Tabuada, Gon\c{c}alo, A universal characterization of higher algebraic \(K\)-theory, Geom. Topol., 17, 2, 733-838 (2013) · Zbl 1267.19001
[12] Braunling, O.; Groechenig, M.; Wolfson, J., Geometric and analytic structures on the higher ad\`eles, Res. Math. Sci., 3, Paper No. 22, 56 pp. (2016) · Zbl 1405.11148
[13] Braunling, O.; Groechenig, M.; Wolfson, J., Operator ideals in Tate objects, Math. Res. Lett., 23, 6, 1565-1631 (2016) · Zbl 1375.18062
[14] Braunling, Oliver; Groechenig, Michael; Wolfson, Jesse, Tate objects in exact categories, Mosc. Math. J., 16, 3, 433-504 (2016) · Zbl 1386.18036
[15] Braunling, Oliver; Groechenig, Michael; Wolfson, Jesse, The \(A_\infty \)-structure of the index map, Ann. K-Theory, 3, 4, 581-614 (2018) · Zbl 1408.19003
[16] Braunling, Oliver; Groechenig, Michael; Wolfson, Jesse, The index map in algebraic \(K\)-theory, Selecta Math. (N.S.), 24, 2, 1039-1091 (2018) · Zbl 1391.19005
[17] Braunling, Oliver, On the local residue symbol in the style of Tate and Beilinson, New York J. Math., 24, 458-513 (2018) · Zbl 1398.32004
[18] Brown, Kenneth S., Cohomology of groups, Graduate Texts in Mathematics 87, x+306 pp. (1982), Springer-Verlag, New York-Berlin · Zbl 0584.20036
[19] Contou-Carr\`ere, Carlos E., La jacobienne g\'{e}n\'{e}ralis\'{e}e d’une courbe relative; construction et propri\'{e}t\'{e} universelle de factorisation, C. R. Acad. Sci. Paris S\'{e}r. A-B, 289, 3, A203-A206 (1979) · Zbl 0447.14005
[20] Contou-Carr\`ere, Carlos E., Corps de classes local g\'{e}om\'{e}trique relatif, C. R. Acad. Sci. Paris S\'{e}r. I Math., 292, 9, 481-484 (1981) · Zbl 0506.14043
[21] Contou-Carr\`ere, C., Jacobiennes g\'{e}n\'{e}ralis\'{e}es globales relatives. The Grothendieck Festschrift, Vol. II, Progr. Math. 87, 69-109 (1990), Birkh\"{a}user Boston, Boston, MA
[22] Contou-Carr\`ere, Carlos, Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole mod\'{e}r\'{e}, C. R. Acad. Sci. Paris S\'{e}r. I Math., 318, 8, 743-746 (1994) · Zbl 0840.14031
[23] Contou-Carr\`ere, Carlos, Jacobienne locale d’une courbe formelle relative, Rend. Semin. Mat. Univ. Padova, 130, 1-106 (2013) · Zbl 1317.14100
[24] Conrad, Brian, Deligne’s notes on Nagata compactifications, J. Ramanujan Math. Soc., 22, 3, 205-257 (2007) · Zbl 1142.14001
[25] Deligne, P., Le symbole mod\'{e}r\'{e}, Inst. Hautes \'{E}tudes Sci. Publ. Math., 73, 147-181 (1991) · Zbl 0749.14011
[26] Dwyer, W. G.; Greenlees, J. P. C., Complete modules and torsion modules, Amer. J. Math., 124, 1, 199-220 (2002) · Zbl 1017.18008
[27] Drinfeld, Vladimir, Infinite-dimensional vector bundles in algebraic geometry: an introduction. The unity of mathematics, Progr. Math. 244, 263-304 (2006), Birkh\"{a}user Boston, Boston, MA · Zbl 1108.14012
[28] A. I. Efimov, Formal completion of a category along a subcategory, 1006.4721, 06 2010.
[29] Geisser, Thomas, Motivic cohomology, \(K\)-theory and topological cyclic homology. Handbook of \(K\)-theory. Vol. 1, 2, 193-234 (2005), Springer, Berlin · Zbl 1113.14017
[30] Gillet, Henri Antoine, The applications of algebraic K-theory to intersection theory, (no paging) pp. (1978), ProQuest LLC, Ann Arbor, MI
[31] Goerss, Paul G.; Jardine, John F., Simplicial homotopy theory, Modern Birkh\"{a}user Classics, xvi+510 pp. (2009), Birkh\"{a}user Verlag, Basel · Zbl 1195.55001
[32] Greenlees, J. P. C.; May, J. P., Derived functors of \(I\)-adic completion and local homology, J. Algebra, 149, 2, 438-453 (1992) · Zbl 0774.18007
[33] Gorchinski\u{\i }, S. O.; Osipov, D. V., Explicit formula for the higher-dimensional Contou-Carr\`ere symbol, Uspekhi Mat. Nauk. Russian Math. Surveys, 70 70, 1, 171-173 (2015) · Zbl 1328.19003
[34] Gorchinski\u{\i }, S. O.; Osipov, D. V., A higher-dimensional Contou-Carr\`ere symbol: local theory, Mat. Sb.. Sb. Math., 206 206, 9-10, 1191-1259 (2015) · Zbl 1337.19004
[35] Gaitsgory, Dennis; Rozenblyum, Nick, DG indschemes. Perspectives in representation theory, Contemp. Math. 610, 139-251 (2014), Amer. Math. Soc., Providence, RI · Zbl 1316.14006
[36] Grothendieck, A., \'{E}l\'{e}ments de g\'{e}om\'{e}trie alg\'{e}brique. III. \'{E}tude cohomologique des faisceaux coh\'{e}rents. I, Inst. Hautes \'{E}tudes Sci. Publ. Math., 11, 167 pp. (1961)
[37] A. Grothendieck, Letter to J.-P. Serre, August 9th, 1960, Correspondance Grothendieck-Serre, Documents Math\'ematiques (Paris) [Mathematical Documents (Paris)], vol. 2, Soci\'et\'e Math\'ematique de France, Paris, 2001. 1942134
[38] M. Groth, A short course on infinity-categories, 1007.2925, 07 2010.
[39] Hovey, Mark, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra, 165, 1, 63-127 (2001) · Zbl 1008.55006
[40] Hovey, Mark; Shipley, Brooke; Smith, Jeff, Symmetric spectra, J. Amer. Math. Soc., 13, 1, 149-208 (2000) · Zbl 0931.55006
[41] Huber, A., On the Parshin-Be\u{\i }linson ad\`eles for schemes, Abh. Math. Sem. Univ. Hamburg, 61, 249-273 (1991) · Zbl 0763.14006
[42] Ivorra, Florian; R\"{u}lling, Kay, K-groups of reciprocity functors, J. Algebraic Geom., 26, 2, 199-278 (2017) · Zbl 1360.19007
[43] M. M. Kapranov, Semi-infinite symmetric powers, math/0107089v1 [math.QA], 2001.
[44] Kato, Kazuya, A generalization of local class field theory by using \(K\)-groups. I, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 26, 2, 303-376 (1979) · Zbl 0428.12013
[45] Kato, Kazuya, Class field theory and algebraic \(K\)-theory. Algebraic geometry, Tokyo/Kyoto, 1982, Lecture Notes in Math. 1016, 109-126 (1983), Springer, Berlin · Zbl 0544.12010
[46] Kato, Kazuya, 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. 55, 241-253 (1986), Amer. Math. Soc., Providence, RI · Zbl 0603.14009
[47] Kato, Kazuya, Existence theorem for higher local fields. Invitation to higher local fields, M\"{u}nster, 1999, Geom. Topol. Monogr. 3, 165-195 (2000), Geom. Topol. Publ., Coventry · Zbl 1008.11061
[48] Keller, Bernhard, On the cyclic homology of exact categories, J. Pure Appl. Algebra, 136, 1, 1-56 (1999) · Zbl 0923.19004
[49] Kerz, Moritz, The Gersten conjecture for Milnor \(K\)-theory, Invent. Math., 175, 1, 1-33 (2009) · Zbl 1188.19002
[50] Kerz, Moritz, Milnor \(K\)-theory of local rings with finite residue fields, J. Algebraic Geom., 19, 1, 173-191 (2010) · Zbl 1190.14021
[51] Kato, Kazuya; Saito, Shuji, 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. 55, 255-331 (1986), Amer. Math. Soc., Providence, RI · Zbl 0614.14001
[52] Kahn, Bruno; Saito, Shuji; Yamazaki, Takao, Reciprocity sheaves, Compos. Math., 152, 9, 1851-1898 (2016) · Zbl 1419.19001
[53] Kapranov, Mikhail; Vasserot, \'{E}ric, Formal loops. II. A local Riemann-Roch theorem for determinantal gerbes, Ann. Sci. \'{E}cole Norm. Sup. (4), 40, 1, 113-133 (2007) · Zbl 1129.14022
[54] Lamotke, Klaus, Semisimpliziale algebraische Topologie, Die Grundlehren der mathematischen Wissenschaften, Band 147, viii+285 pp. (1968), Springer-Verlag, Berlin-New York · Zbl 0188.28301
[55] Loday, Jean-Louis, \(K\)-th\'{e}orie alg\'{e}brique et repr\'{e}sentations de groupes, Ann. Sci. \'{E}cole Norm. Sup. (4), 9, 3, 309-377 (1976) · Zbl 0362.18014
[56] J. Lurie, Higher algebra. · JFM 44.0127.05
[57] Lurie, Jacob, Higher topos theory, Annals of Mathematics Studies 170, xviii+925 pp. (2009), Princeton University Press, Princeton, NJ · Zbl 1175.18001
[58] J. Lurie, Derived algebraic geometry XII: Proper morphisms, completions, and the Grothendieck existence theorem, arXiv:0911.0018, 11 2009.
[59] J. Lurie, (infinity,2)-categories and the Goodwillie Calculus I.
[60] May, J. Peter, \(E_{\infty }\) ring spaces and \(E_{\infty }\) ring spectra, Lecture Notes in Mathematics, Vol. 577, 268 pp. (1977), Springer-Verlag, Berlin-New York · Zbl 0345.55007
[61] May, J. Peter, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, viii+161 pp. (1992), University of Chicago Press, Chicago, IL · Zbl 0769.55001
[62] Milnor, John, Algebraic \(K\)-theory and quadratic forms, Invent. Math., 9, 318-344 (1969/70) · Zbl 0199.55501
[63] M. Morrow, An introduction to higher dimensional local fields and ad\`eles, 1204.0586.
[64] Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles, Lecture notes on motivic cohomology, Clay Mathematics Monographs 2, xiv+216 pp. (2006), American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA · Zbl 1115.14010
[65] Musicantov, Evgeny; Yom Din, Alexander, Reciprocity laws and \(K\)-theory, Ann. K-Theory, 2, 1, 27-46 (2017) · Zbl 1354.19003
[66] Osipov, D. V., Central extensions and reciprocity laws on algebraic surfaces, Mat. Sb.. Sb. Math., 196 196, 9-10, 1503-1527 (2005) · Zbl 1177.14083
[67] Osipov, Denis; Zhu, Xinwen, A categorical proof of the Parshin reciprocity laws on algebraic surfaces, Algebra Number Theory, 5, 3, 289-337 (2011) · Zbl 1237.19007
[68] Osipov, Denis; Zhu, Xinwen, The two-dimensional Contou-Carr\`ere symbol and reciprocity laws, J. Algebraic Geom., 25, 4, 703-774 (2016) · Zbl 1346.19003
[69] P\'{a}l, Ambrus, On the kernel and the image of the rigid analytic regulator in positive characteristic, Publ. Res. Inst. Math. Sci., 46, 2, 255-288 (2010) · Zbl 1202.19005
[70] Par\v{s}in, A. N., Abelian coverings of arithmetic schemes, Dokl. Akad. Nauk SSSR, 243, 4, 855-858 (1978) · Zbl 0443.12006
[71] Parshin, A. N., Local class field theory, Trudy Mat. Inst. Steklov., 165, 143-170 (1984) · Zbl 0535.12013
[72] A. Parshin and T. Fimmel, Introduction to higher adelic theory (draft), unpublished, 1999.
[73] Pablos Romo, Fernando, On the tame symbol of an algebraic curve, Comm. Algebra, 30, 9, 4349-4368 (2002) · Zbl 1055.14017
[74] Previdi, Luigi, Locally compact objects in exact categories, Internat. J. Math., 22, 12, 1787-1821 (2011) · Zbl 1255.18012
[75] Porta, Marco; Shaul, Liran; Yekutieli, Amnon, Completion by derived double centralizer, Algebr. Represent. Theory, 17, 2, 481-494 (2014) · Zbl 1332.13015
[76] 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. · Zbl 1330.13022
[77] Quillen, Daniel, Higher algebraic \(K\)-theory. I. Algebraic \(K\)-theory, I: Higher \(K\)-theories, Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972, 85-147. Lecture Notes in Math., Vol. 341 (1973), Springer, Berlin · Zbl 0292.18004
[78] Safronov, Pavel, Quasi-Hamiltonian reduction via classical Chern-Simons theory, Adv. Math., 287, 733-773 (2016) · Zbl 1440.53096
[79] Saito, Sho, On Previdi’s delooping conjecture for \(K\)-theory, Algebra Number Theory, 9, 1, 1-11 (2015) · Zbl 1350.19001
[80] Schenzel, Peter, Proregular sequences, local cohomology, and completion, Math. Scand., 92, 2, 161-180 (2003) · Zbl 1023.13011
[81] Schlichting, Marco, Delooping the \(K\)-theory of exact categories, Topology, 43, 5, 1089-1103 (2004) · Zbl 1059.18007
[82] Schlichting, Marco, Negative \(K\)-theory of derived categories, Math. Z., 253, 1, 97-134 (2006) · Zbl 1090.19002
[83] Serre, Jean-Pierre, Algebraic groups and class fields, Graduate Texts in Mathematics 117, x+207 pp. (1988), Springer-Verlag, New York · Zbl 0703.14001
[84] Tate, John, Residues of differentials on curves, Ann. Sci. \'{E}cole Norm. Sup. (4), 1, 149-159 (1968) · Zbl 0159.22702
[85] The Stacks Project Authors, Stacks project, http://math.columbia.edu/algebraic_geometry/stacks-git.
[86] Thomason, R. W.; Trobaugh, Thomas, Higher algebraic \(K\)-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, Progr. Math. 88, 247-435 (1990), Birkh\"{a}user Boston, Boston, MA
[87] Waldhausen, Friedhelm, Algebraic \(K\)-theory of spaces. Algebraic and geometric topology, New Brunswick, N.J., 1983, Lecture Notes in Math. 1126, 318-419 (1985), Springer, Berlin · Zbl 0579.18006
[88] Weibel, Charles A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, xiv+450 pp. (1994), Cambridge University Press, Cambridge · Zbl 0797.18001
[89] Yekutieli, Amnon, An explicit construction of the Grothendieck residue complex, Ast\'{e}risque, 208, 127 pp. (1992) · Zbl 0788.14011
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