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The two-dimensional Contou-Carrère symbol and reciprocity laws. (English) Zbl 1346.19003
In a previous article [Algebra Number Theory 5, No. 3, 289–337 (2011; Zbl 1237.19007)] the authors of this paper gave a new conceptual proof of Parshin-type reciprocity laws for algebraic surfaces. They used the machinery of categorical central extensions and generalized commutators to construct the appropriate two-dimensional tame symbols needed for the formulation of these reciprocity laws. The purpose of the current paper is to extend the results of that paper from a ground field \(k\) to a general commutative ground ring \(R\).
The notion of central extension is extended to the situation of more general sheaves on a topos and the \(2\)-Tate vector spaces of the previous article are replaced with Drinfeld’s notion of a Tate \(R\)-module. In this way, analogously to the previous article, from the existence of a central extension of \(\mathcal{P}ic^{\mathbb{Z}}\) by \(L^2\mathbb{G}_m\) is deduced a tri-multiplicative map of functors (the two-dimensional Contou-Carrére symbol) \[ C_3: L^2\mathbb{G}_m\times L^2\mathbb{G}_m\times L^2\mathbb{G}_m\to \mathbb{G}_m. \] Here \(\mathcal{P}ic^{\mathbb{Z}}\) is the Picard groupoid of graded lines and \(L^2\mathbb{G}_m\) is the group-valued functor \(L^2\mathbb{G}_m(R)=R((u))((t))^*\) for a ring \(R\). With this construction, the authors show that \(C_3\) is invariant under change of local parameters \((u,t)\to (u',t')\).
In section \(6\) of the paper the authors prove the following two-dimensional reciprocity laws (Theorem 6.1): Let \(X\) be a smooth algebraic surface over a perfect field \(k\) and let \(R\) be a local finite \(k\)-algebra. Then
For any closed point \(x\in X\) and for any \(f,g,h\in (K_x\otimes_kR)^*\) we have \[ \prod_{C\ni x}(f,g,h)_{x,C}=1, \] the product being taken over all irreducible curves \(C\) containing \(x\)
Let \(C\) be a projective irreducible curve on \(X\). Then for any \(f,g,h\in (K_C\otimes_kR)^*\) we have \[ \prod_{x\in C}\mathrm{Nm}_{k(x)/k}(f,g,h)_{x,C}=1, \] where the product is taken over all closed points \(x\in C\).
Here the elements \((f,g,h)_{x,C}\in (k(x)\otimes R)^*\) are defined using the map \(C_3\).
In the final section of the paper, the authors use the Contou-Carrére symbol over the ring \(R=\mathbb{F}_q[s]/s^{n+1}\) to obtain Parshin’s local symbol which is used to construct the generalized Artin-Schreier-Witt duality for the two-dimensional local field \(\mathbb{F}_q((u))((t))\).
A considerable portion of the paper, however, is devoted to obtaining explicit descriptions of the Contou-Carrére symbol analogous the much simpler one-dimensional case. In particular, in section 3 the authors show that when \(\mathbb{Q}\subset R\) and for appropriate \(f,g,h\in R((u))((t))^*\) we have \[ C_3(f,g,h)=\exp\mathrm{Res}\left(\log f \frac{dg}{g}\wedge \frac{dh}{h}\right) \] In section 7, the authors discuss a \(K\)-theoretic definition of the Contou-Carrére symbols from which the first of the above reciprocity laws also follows naturally.

19D45 Higher symbols, Milnor \(K\)-theory
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[1] Anderson, Greg W.; Pablos Romo, Fernando, Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an Artinian local ring, Comm. Algebra, 32, 1, 79-102, (2004) · Zbl 1077.14033
[2] Beilinson, Alexander; Bloch, Spencer; Esnault, H\'el\`ene, \(ϵ \)-factors for Gauss-Manin determinants, Dedicated to Yuri I. Manin on the occasion of his 65th birthday, Mosc. Math. J., 2, 3, 477-532, (2002) · Zbl 1061.14010
[3] Breen, Lawrence, Bitorseurs et cohomologie non ab\'elienne. The Grothendieck Festschrift, Vol. I, Progr. Math. 86, 401-476, (1990), Birkh\"auser Boston, Boston, MA · Zbl 0743.14034
[4] Breen, Lawrence, On the classification of \(2\)-gerbes and \(2\)-stacks, Ast\'erisque, 225, 160 pp., (1994) · Zbl 0818.18005
[5] Brylinski, J.-L.; McLaughlin, D. A., The geometry of two-dimensional symbols, \(K\)-Theory, 10, 3, 215-237, (1996) · Zbl 0870.32004
[6] Contou-Carr\`ere, Carlos, Jacobienne locale, groupe de bivecteurs de Witt universel, et symbole mod\'er\'e, C. R. Acad. Sci. Paris S\'er. I Math., 318, 8, 743-746, (1994) · Zbl 0840.14031
[7] Contou-Carr\`ere, Carlos, Jacobienne locale d’une courbe formelle relative, Rend. Semin. Mat. Univ. Padova, 130, 1-106, (2013) · Zbl 1317.14100
[8] Deligne, P., Le symbole mod\'er\'e, Inst. Hautes \'Etudes Sci. Publ. Math., 73, 147-181, (1991) · Zbl 0749.14011
[9] Drinfeld, Vladimir, Infinite-dimensional vector bundles in algebraic geometry: an introduction. The unity of mathematics, Progr. Math. 244, 263-304, (2006), Birkh\"auser Boston, Boston, MA · Zbl 1108.14012
[10] Frenkel, Edward; Zhu, Xinwen, Gerbal representations of double loop groups, Int. Math. Res. Not. IMRN, 17, 3929-4013, (2012) · Zbl 1280.22024
[11] [G]G D. Gaitsgory, <span class=”textit”>A</span>ffine Grassmannian and the loop group, Seminar Notes written by D. Gaitsgory and N. Rozenblyum, 2009, 12 pp., available at http://\linebreak www.math.harvard.edu/\(~ \)gaitsgde/grad_2009/SeminarNotes/Oct13(AffGr).pdf
[12] Grayson, Daniel, Higher algebraic \(K\)-theory. II (after Daniel Quillen). Algebraic \(K\)-theory (Proc. Conf., Northwestern Univ., Evanston, Ill., 1976), 217-240, (1976), Lecture Notes in Math., Vol. 551, Springer, Berlin
[13] [HL]H I. Horozov and Zh. Luo, <span class=”textit”>O</span>n the Contou-Carrere symbol for surfaces, preprint, arXiv:1310.7065 [math.AG].
[14] [Kap]Kap M. Kapranov, <span class=”textit”>S</span>emiinfinite symmetric powers, preprint, arXiv:math/0107089 [math.QA].
[15] Kapranov, Mikhail; Vasserot, \'Eric, Formal loops. II. A local Riemann-Roch theorem for determinantal gerbes, Ann. Sci. \'Ecole Norm. Sup. (4), 40, 1, 113-133, (2007) · Zbl 1129.14022
[16] Kato, Kazuya, A generalization of local class field theory by using \(K\)-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27, 3, 603-683, (1980) · Zbl 0463.12006
[17] Kato, Kazuya, Generalized class field theory. Proceedings of the International Congress of Mathematicians, Vol. I, II, Kyoto, 1990, 419-428, (1991), Math. Soc. Japan, Tokyo · Zbl 0827.11073
[18] Kato, Kazuya; Russell, Henrik, Modulus of a rational map into a commutative algebraic group, Kyoto J. Math., 50, 3, 607-622, (2010) · Zbl 1206.14069
[19] Kato, Kazuya; Saito, Shuji, Two-dimensional class field theory. Galois groups and their representations, Nagoya, 1981, Adv. Stud. Pure Math. 2, 103-152, (1983), North-Holland, Amsterdam
[20] Morava, Jack, An algebraic analog of the Virasoro group, Quantum groups and integrable systems (Prague, 2001), Czechoslovak J. Phys., 51, 12, 1395-1400, (2001) · Zbl 1045.22022
[21] Osipov, D. V., Adelic constructions of direct images of differentials and symbols, Mat. Sb.. Sb. Math., 188 188, 5, 697-723, (1997) · Zbl 0909.14011
[22] 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
[23] Osipov, Denis, Adeles on \(n\)-dimensional schemes and categories \(C_n\), Internat. J. Math., 18, 3, 269-279, (2007) · Zbl 1126.14004
[24] Osipov, Denis V., \(n\)-dimensional local fields and adeles on \(n\)-dimensional schemes. Surveys in contemporary mathematics, London Math. Soc. Lecture Note Ser. 347, 131-164, (2008), Cambridge Univ. Press, Cambridge · Zbl 1144.11078
[25] 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
[26] P\'al, 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
[27] Par\vsin, A. N., Class fields and algebraic \(K\)-theory, Uspehi Mat. Nauk, 30, 1 (181), 253-254, (1975)
[28] Par\vsin, A. N., Abelian coverings of arithmetic schemes, Dokl. Akad. Nauk SSSR, 243, 4, 855-858, (1978)
[29] Parshin, A. N., Local class field theory, Algebraic geometry and its applications, Trudy Mat. Inst. Steklov., 165, 143-170, (1984) · Zbl 0535.12013
[30] Srinivas, V., Algebraic \(K\)-theory, Progress in Mathematics 90, xviii+341 pp., (1996), Birkh\"auser Boston, Inc., Boston, MA · Zbl 0860.19001
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