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Higher-dimensional Contou-Carrère symbol and continuous automorphisms. (English. Russian original) Zbl 1360.19005
Funct. Anal. Appl. 50, No. 4, 268-280 (2016); translation from Funkts. Anal. Prilozh. 50, No. 4, 26-42 (2016).
For a natural number \(n\) and an ring \(A\) let \(\mathcal{L}^n(A)\) be the algebra of iterated Laurent series \(A((t_1))\ldots ((t_1))\). The \(n\) dimensional Contou-Carrère symbol is an anti-symmetric multi-linear map \[ CC_n:(\mathcal{L}(A)^*)^{\times (n+1)}\rightarrow A^*\,. \]
In this paper it is shown that for a continuous endomorphism \(\phi:\mathcal{L}^n(A)\rightarrow \mathcal{L}^n(A)\) we have for all \(f_1,\ldots, f_{n+1}\in \mathcal{L}^n(A)\) that \[ CC_n(\phi(f_1),\ldots, \phi(f_{n+1}))= CC_n(f_1,\ldots, f_{n+1})^{d(\phi)}\,, \] where \(d(\phi)\) is the determinant of a certain associated matrix. In particular, \(d(\phi)=1\) if \(\phi\) is an automorphism.This is generalised to continuous homomorphisms from \(\mathcal{L}^n(A)\) to \(\mathcal{L}^m(A)\) and an example is given to show it does not extend to non-continuous automorphisms. The above invariance for continuous automorphisms is then used to give an explicit, fairly elementary, formula for the higher-dimensional Contou-Carrère symbol for any ring \(A\).

19D45 Higher symbols, Milnor \(K\)-theory
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
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[1] Anderson, G.; Pablos Romo, F., Simple proofs of classical explicit reciprocity laws on curves using determinant groupoids over an Artinian local ring, Comm. Algebra, 32, 79-102, (2004) · Zbl 1077.14033
[2] Contou-Carrère, C., Jacobienne locale, groupe de bivecteurs De Witt universel, et symbole modéré, C. R. Acad. Sci. Paris Sér. I Math., 318, 743-746, (1994) · Zbl 0840.14031
[3] Contou-Carrère, C., Jacobienne locale d’une courbe formelle relative, Rend. Semin. Mat. Univ. Padova, 130, 1-106, (2013) · Zbl 1317.14100
[4] Deligne, P., Le symbole modéré, Publ. Math. IHES, 73, 147-181, (1991) · Zbl 0749.14011
[5] Gorchinskiy, S. O.; Osipov, D. V., Explicit formula for the higher-dimensional contou-carrère symbol, Uspekhi Mat. Nauk, 68, 183-184, (2015) · Zbl 1328.19003
[6] Gorchinskiy, S. O.; Osipov, D. V., Tangent space to Milnor K-groups of rings, Trudy Mat. Inst. Steklov., 290, 34-42, (2015) · Zbl 1334.19003
[7] Gorchinskiy, S. O.; Osipov, D. V., A higher-dimensional contou-carrère symbol: local theory, Mat. Sb., 206, 21-98, (2015) · Zbl 1337.19004
[8] Gorchinskiy, S. O.; Osipov, D. V., Continuous homomorphisms between algebras of iterated Laurent series over a ring, TrudyMat. Inst. Steklov., 294, 54-75, (2016) · Zbl 1359.13023
[9] Grayson, D., Higher algebraic K-theory. II (after daniel Quillen), 217-240, (1976)
[10] Kato, K., A generalization of local class field theory by using K-groups. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27, 603-683, (1980) · Zbl 0463.12006
[11] Osipov, D.; Zhu, X., The two-dimensional contou-carrère symbol and reciprocity laws, J. Algebraic Geom., 25, 703-774, (2016) · Zbl 1346.19003
[12] Parshin, A. N., Local class field theory, Algebraic geometry and its applications. Trudy Mat. Inst. Steklov., 165, 143-170, (1984) · Zbl 0535.12013
[13] A. Yekutieli, “An explicit construction of the Grothendieck residue complex. With an appendix by P. Sastry,” Astérisque, 208 (1992). · Zbl 0788.14011
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