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].

MSC:

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

Keywords:

Contou-Carrére symbol; $$K$$-theory
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