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Explicit formula for the higher-dimensional Contou-Carrère symbol. (English. Russian original) Zbl 1328.19003
Russ. Math. Surv. 70, No. 1, 171-173 (2015); translation from Usp. Mat. Nauk. 70, No. 1, 183-184 (2015).
Let \(A\) be a commutative associative unital ring. For \(n\geq 1\), let \(L_n(A)\) denote the iterated Laurent series ring \(A((t_1))\dots ((t_n))\). 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: The \(f_i\) can be considered as elements of \(K_1(L_n(A))\) via the canonical embedding \(R^*\to K_1(R)\) for commutative rings \(R\). Let \(\{ f_1,\ldots, f_{n+1}\}\) denote the corresponding product in \(K_{n+1}(L_n(A))\). Let \(\partial_m\) denote the canonical map \(K_m(R((t)))\to K_{m-1}(R)\) for a commutative ring \(R\) and let \(\mathrm{det}\) be the natural 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 \(\gamma_{n+1}:= \mathrm{det}\circ \partial_2\circ \cdots \circ \partial_{n+1}\).
This symbol generalizes the classical Contou-Carrère pairing \(CC_1\) and can be used to construct higher local pairings in Parshin-Kato \(n\)-dimensional class field theory.
In the present note the authors announce an explicit formula for \(CC_n\) is the case where \(A\) is a \(\mathbb{Q}\)-algebra. This generalizes the case \(n=2\) which was already studied by the second author and X. Zhu [“The two-dimensional Contou-Carrére symbol and reciprocity laws”, Preprint, arxiv:1305.6032]. The details of the proof are provided elsewhere by the authors see [“Higher Contou-Carrére symbol: local theory”, Preprint, arxiv:1505.03829].

19D45 Higher symbols, Milnor \(K\)-theory
19F15 Symbols and arithmetic (\(K\)-theoretic aspects)
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