zbMATH — the first resource for mathematics

A higher-dimensional Contou-Carrère symbol: local theory. (English. Russian original) Zbl 1337.19004
Sb. Math. 206, No. 9, 1191-1259 (2015); translation from Mat. Sb. 206, No. 9, 21-98 (2015).
For any (unital, associative, commutative) ring \(A\) let \(L_n(A)\) denote the iterated Laurent series ring \(A((t_1))\cdots ((t_n))\). The authors define the \(n\)-dimensional Contou-Carrère symbol to be the map \[ CC_n: (L_n(A)^*)^{n+1} \to A^* \] defined as follows:
\(L_n(A)^*\) is a subgroup of \(K_1(L_n(A))\) via the canonical embedding \(R^*\to K_1(R)\) for commutative rings \(R\). Given \(f_i\in L_n(A)^*\), \(1\leq i\leq n+1\), let \(\{ f_1,\ldots, f_{n+1}\}\) denote the corresponding product in \(K_{n+1}(L_n(A))\). There are boundary homomorphisms \(\partial_m:K_m(R((t)))\to K_{m-1}(R)\) for a commutative ring \(R\). Let \(\mathrm{det}\) be the determinant map \(K_1(A)\to A^*\). Then \[ CC_n(f_1,\ldots, f_{n+1}):= \gamma_{n+1}(\{ f_1,\ldots, f_{n+1}\}) \] where \(\gamma_{n+1}:= \mathrm{det}\circ \partial_2\circ \cdots \circ \partial_{n+1}\).
The main results of the article under review are to show that the map so-defined agrees with the original Contou-Carrère symbol in the case \(n=1\) [C. Contou-Carrère, C. R. Acad. Sci., Paris, Sér. I 318, No. 8, 743–746 (1994; Zbl 0840.14031)] and with its extension to dimension \(n=2\) by the second author and X. Zhu [“The two-dimensional Contou-Carrère symbol and reciprocity laws”, to appear in J. Algebraic Geom., arXiv: 1305.6032]. The well-written introduction to the article gives some details of the background and history of the Contou-Carrère symbol.
Furthermore, the authors prove an explicit formula for \(CC_n\) in the case \(A\) is a \(\mathbb{Q}\)-algebra. In particular, \[ CC_n(f_1,\ldots, f_{n+1})=\exp\mathrm{res}\left(\log(f_1)\frac{df_2}{f_2}\wedge \cdots \wedge \frac{df_{n+1}}{f_{n+1}}\right) \] when \(f_1\) satisfies additional conditions to ensure that \(\log(f_1)\) is well-defined.
In the final section of the article, the authors use the symbol \(CC_n\) to give an explicit description of A. N. Parshin’s local reciprocity map in local class field theory for \(n\)-dimensional local fields of characteristic \(p>0\) [Proc. Steklov Inst. Math. 165, 157–185 (1985; Zbl 0579.12012)]. In particular, they use it to define a higher-dimensional Witt pairing.
The key result in the paper is a uniqueness theorem, characterising the symbol \(CC_n\): Given a functor \(F\) from rings to abelian groups let \(LF\) be the functor \((LF)(A):= F(A((t)))\). Thus \(L^n\mathbb{G}_m(A)=A((t_1))\cdots ((t_n))^*=L_n(A)^*\). By construction, the symbol \(CC_n\) factors as a map of functors \[ (L^n\mathbb{G}_m)^{\times n+1} \to L^nK_{n+1}^M\to \mathbb{G}_m \] where, for a ring \(B\), \(K_n^M(B)\) is the quotient of \((B^*)^{\otimes n}\) by the subgroup generated by the elements of the form \(f_1\otimes \cdots \otimes f_n\) where \(f_{i+1}=1-f_i\) for some \(1\leq i\leq n-1\). Here the second map is a morphism of group-valued functors; in particular, the authors prove in the article that the \(K\)-theory boundary maps \(\partial_m\) are maps of functors. The authors prove (Theorem 8.10) that the group \(\mathrm{Hom}^{\mathrm{gr}}(L^nK^M_{n+1},\mathbb{G}_m)\) of maps of group-valued functors is infinite cyclic with generator \(CC_n\).
The authors’ method of proof of their main theorem is of interest in its own right. The iterated loop groups \(L^n\mathbb{G}_m\), and certain special subfunctors, are naturally represented by inductive systems of affine schemes, or ind-affine schemes. The authors introduce the class of thick ind-cones and develop their properties in order to prove the main theorems.

19D45 Higher symbols, Milnor \(K\)-theory
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
Full Text: DOI arXiv
[1] C. Contou-Carre\gravere 1994 Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole mode\acutere\acute C. R. Acad. Sci. Paris Se\acuter. I Math.318 8 743–746
[2] C. Contou-Carre\gravere 2013 Jacobienne locale d’une courbe formelle relative Rend. Semin. Mat. Univ. Padova130 1–106 · Zbl 1317.14100 · doi:10.4171/RSMUP/130-1
[3] A. A. Beilinson 1980 Higher regulators and values of L-functions of curves Funktsional. Anal. i Prilozhen.14 2 46–47
[4] English transl. in A. A. Beilinson 1980 Funct. Anal. Appl.14 2 116–118 · Zbl 0475.14015 · doi:10.1007/BF01086554
[5] P. Deligne 1991 Le symbole mode\acutere\acute Inst. Hautes E\acutetudes Sci. Publ. Math.73 147–181
[6] H. Esnault and E. Viehweg 1988 Deligne-Beilinson cohomology Beilinson’s conjectures on special values of L-functions Perspect. Math. 4 Academic Press, Boston, MA 43–91
[7] D. Osipov and Xinwen Zhu 1988 The two-dimensional Contou-Carre\gravere symbol and reciprocity laws J. Algebraic Geom.
[8] D. Osipov and Xinwen Zhu 1988 1305.6032v2
[9] G. W. Anderson and F. Pablos Romo 2004 Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an Artinian local ring Comm. Algebra32 1 79–102 · Zbl 1077.14033 · doi:10.1081/AGB-120027853
[10] S. M. Gersten 1974 The localization theorem for projective modules Comm. Algebra2 4 317–350 · Zbl 0332.18013 · doi:10.1080/00927877408822015
[11] D. Grayson 1976 Higher algebraic K-theory. II (after Daniel Quillen) Algebraic K-theoryNorthwestern Univ., Evanston, Ill. 1976 Lecture Notes in Math. 551 Springer-Verlag, Berlin–New York 217–240
[12] K. Kato 1980 A generalization of local class field theory by using K-groups. II J. Fac. Sci. Univ. Tokyo Sect. IA Math.27 3 603–683 · Zbl 0463.12006
[13] J.-L. Brylinski and D. A. McLaughlin 1996 Mulitdimenisonal reciprocity laws J. Reine Angew. Math.481 125–147
[14] S. O. Gorchinskiy and D. V. Osipov 2015 Explicit formula for the higher-dimensional Contou-Carre\gravere symbol Uspekhi Mat. Nauk70 1(421) 183–184 · doi:10.4213/rm9653
[15] English transl. in S. O. Gorchinskiy and D. V. Osipov 2015 Russian Math. Surveys70 1 171–173
[16] A. N. Parshin 1984 Local class field theory Algebraic geometry and its applications, Collection of articles Trudy Mat. Inst. Steklov. 165 143–170
[17] English transl. in A. N. Parshin 1985 Proc. Steklov Inst. Math.165 157–185
[18] S. O. Gorchinskiy and D. V. Osipov 2015 Tangent space to Milnor K-groups of rings Trudy Mat. Inst. Steklov.290 34–42 · doi:10.1134/S0371968515030036
[19] English transl. in S. O. Gorchinskiy and D. V. Osipov 2015 Proc. Steklov Inst. Math.290 26–34
[20] S. O. Gorchinskiy and D. V. Osipov 2015 1505.03780v1
[21] S. Bloch 1973 On the tangent space to Quillen K-theory Algebraic K-theoryBattelle Memorial Inst., Seattle, Wash. 1972 Lecture Notes in Math. 341, I Higher K-theories Springer-Verlag, Berlin–New York 205–210 · doi:10.1007/BFb0067058
[22] M. Kontsevich 1973 Noncommutative identities 1109.2469v1
[23] Z. I. Borevich and I. R. Shafarevich 1964 Number theory Nauka, Moscow 566 pp. · Zbl 0121.04202
[24] English transl. Z. I. Borevich and I. R. Shafarevich 1966 Pure and Applied Mathematics 20 Academic Press, New York–London x+435 pp.
[25] S. V. Vostokov, I. B. Zhukov and I. B. Fesenko 1990 On the theory of multidimensional local fields. Methods and constructions Algebra i Analiz2 4 91–118
[26] English transl. in S. V. Vostokov, I. B. Zhukov and I. B. Fesenko 1991 Leningrad Math. J.2 4 775–800
[27] A. Grothendieck and J. L. Verdier 1972 Pre\acutefaisceaux The\acuteorie des topos et cohomologie e\acutetale des sche\acutemas. Tome 1: The\acuteorie des topos. (French), Se\acuteminaire de ge\acuteome\acutetrie alge\acutebrique du Bois-Marie 1963–1964 (SGA 4) Lecture Notes in Math. 269 Springer-Verlag, Berlin–New York 1–217 · doi:10.1007/BFb0081552
[28] M. Artin and B. Mazur 1969 Etale homotopy Lecture Notes in Math. 100 Springer-Verlag, Berlin–New York iii+169 pp. · Zbl 0182.26001 · doi:10.1007/BFb0080957
[29] G. Pappas and M. Rappoport 2008 Twisted loop groups and their affine flag varieties Adv. Math.219 1 118–198 · Zbl 1159.22010 · doi:10.1016/j.aim.2008.04.006
[30] J.-L. Loday 1976 K-the\acuteorie alge\acutebrique et repre\acutesentations de groupes Ann. Sci. E\acutecole Norm. Sup. (4)9 3 309–377
[31] V. Srinivas 1996 Algebraic K-theory Progr. Math. 90 Birkha\ddotuser Boston, Inc., Boston, MA 2nd ed., xviii+341 pp. · doi:10.1007/978-0-8176-4739-1
[32] M. Morrow 2014 K_2 of localisations of local rings J. Algebra399 190–204 · Zbl 1308.19003 · doi:10.1016/j.jalgebra.2013.09.026
[33] Yu. P. Nesterenko and A. A. Suslin 1989 Homology of the full linear group over a local ring, and Milnor’s K-theory Izv. Akad. Nauk SSSR Ser. Mat.53 1 121–146
[34] English transl. in Yu. P. Nesterenko and A. A. Suslin 1990 Math. USSR-Izv.34 1 121–145
[35] M. Kerz 2009 The Gersten conjecture for Milnor K-theory Invent. Math.175 1 1–33 · Zbl 1188.19002 · doi:10.1007/s00222-008-0144-8
[36] R. W. Thomason and T. Trobaugh 1990 Higher algebraic K-theory of schemes and of derived categories The Grothendieck Festschrift Progr. Math. 88, III Birkha\ddotuser Boston, Boston, MA 247–435 · doi:10.1007/978-0-8176-4576-2_10
[37] E. Musicantov and A. Yom Din 1990 Reciprocity laws and K-theory 1410.5391
[38] D. Quillen 1973 Higher algebraic K-theory. I Algebraic K-theoryBattelle Memorial Inst., Seattle, Wash. 1972 Lecture Notes in Math. 341, I Higher K-theories Springer-Verlag, Berlin–New York 85–147 · Zbl 0292.18004 · doi:10.1007/BFb0067053
[39] D. Gaitsgory 2009 Affine Grassmannian and the loop group, Seminar notes written by D. Gaitsgory and N. Rozenblyum 12 pp. http://www.math.harvard.edu/ gaitsgde/grad_2009/SeminarNotes/Oct13(AffGr).pdf
[40] A. A. Beilinson, S. Bloch and H. Esnault 2002 \epsilon-factors for Gauss-Manin determinants Mosc. Math. J.2 3 477–532 · Zbl 1061.14010
[41] O. Braunling, M. Groechenig and J. Wolfson 2002 A generalized Contou-Carre\gravere symbol and its reciprocity laws in higher dimensions 1410.3451v2
[42] O. Braunling, M. Groechenig and J. Wolfson 2002 The index map in algebraic K-theory 1410.1466v2 · Zbl 1375.18062
[43] A. N. Parshin 1990 Galois cohomology and the Brauer group of local fields Galois theory, rings, algebraic groups and their applications, Collected papers Trudy Mat. Inst. Steklov. 183 Nauka, Leningrad 159–169
[44] English transl. in A. N. Parshin 1991 Proc. Steklov Inst. Math.183 191–201
[45] A. G. Khovanskii 2006 An Analog of Determinant Related to Parshin-Kato Theory and Integer Polytopes Funktsional. Anal. i Prilozhen.40 2 55–64 · doi:10.4213/faa6
[46] English transl. in A. G. Khovanskii 2006 Funct. Anal. Appl.40 2 126–133 · Zbl 1109.14036 · doi:10.1007/s10688-006-0019-y
[47] D. Osipov 2003 To the multidimensional tame symbol, preprint 03-13 Humboldt University, Berlin 27 pp. http://edoc.hu-berlin.de/docviews/abstract.php?id=26204
[48] D. Osipov 2003 1105.1362v1
[49] D. V. Osipov 1997 Adele constructions of direct images of differentials and symbols Mat. Sb.188 5 59–84 · doi:10.4213/sm230
[50] English transl. in D. V. Osipov 1997 Sb. Math.188 5 697–723 · Zbl 0909.14011 · doi:10.1070/sm1997v188n05ABEH000230
[51] K. Kato 1983 Residue homomorphisms in Milnor K-theory Galois groups and their representationsNagoya 1981 Adv. Stud. Pure Math. 2 North-Holland, Amsterdam 153–172
[52] M. Asakura 1983 On dlog image of K_2 of elliptic surface minus singular fibers math/0511190
[53] A. Pa\acutel 2010 On the kernel and the image of the rigid analytic regulator in positive characteristic Publ. Res. Inst. Math. Sci.46 2 255–288 · Zbl 1202.19005 · doi:10.2977/PRIMS/9
[54] Dongwen Liu 2014 Kato’s residue homomorphisms and reciprocity laws on arithmetic surfaces Adv. Math.251 1–21 · Zbl 1295.14031 · doi:10.1016/j.aim.2013.10.007
[55] T. Chinburg, G. Pappas and M. J. Taylor 2014 Higher adeles and non-abelian Riemann-Roch 1204.4520v4
[56] K. Kato 1979 A generalization of local class field theory by using K-groups. I J. Fac. Sci. Univ. Tokyo Sect. IA Math.26 2 303–376 · Zbl 0428.12013
[57] I. B. Fesenko 2001 Sequential topologies and quotients of Milnor K-groups of higher local fields Algebra i Analiz13 3 198–221
[58] English transl. in I. B. Fesenko 2002 St. Petersburg Math. J.13 3 485–501
[59] E. Witt 1936 Zyklische Körper und Algebren der Charakteristik p vom Grad p^n. Struktur diskret bewerteter perfekter Körper mit vollkommenem Restklassenkörper der Charakteristik pJ. Reine Angew. Math.176 126–140 · JFM 62.1112.03
[60] D. Mumford 1966 Lectures on curves on an algebraic surface Ann. of Math. Stud. 59 Princeton Univ. Press, Princeton, N.J. xi+200 pp.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.