×

zbMATH — the first resource for mathematics

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 \(A\).

MSC:
19D45 Higher symbols, Milnor \(K\)-theory
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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
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.