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A higher-dimensional Contou-Carrère symbol: local theory. (English. Russian original) Zbl 1337.19004
Sb. Math. 206, No. 9, 1191-1259 (2015); translation from Mat. Sb. 206, No. 9, 21-98 (2015).
For any (unital, associative, commutative) ring \(A\) let \(L_n(A)\) denote the iterated Laurent series ring \(A((t_1))\cdots ((t_n))\). 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:
\(L_n(A)^*\) is a subgroup of \(K_1(L_n(A))\) via the canonical embedding \(R^*\to K_1(R)\) for commutative rings \(R\). Given \(f_i\in L_n(A)^*\), \(1\leq i\leq n+1\), let \(\{ f_1,\ldots, f_{n+1}\}\) denote the corresponding product in \(K_{n+1}(L_n(A))\). There are boundary homomorphisms \(\partial_m:K_m(R((t)))\to K_{m-1}(R)\) for a commutative ring \(R\). Let \(\mathrm{det}\) be the determinant 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 \(\gamma_{n+1}:= \mathrm{det}\circ \partial_2\circ \cdots \circ \partial_{n+1}\).
The main results of the article under review are to show that the map so-defined agrees with the original Contou-Carrère symbol in the case \(n=1\) [C. Contou-Carrère, C. R. Acad. Sci., Paris, Sér. I 318, No. 8, 743–746 (1994; Zbl 0840.14031)] and with its extension to dimension \(n=2\) by the second author and X. Zhu [“The two-dimensional Contou-Carrère symbol and reciprocity laws”, to appear in J. Algebraic Geom., arXiv: 1305.6032]. The well-written introduction to the article gives some details of the background and history of the Contou-Carrère symbol.
Furthermore, the authors prove an explicit formula for \(CC_n\) in the case \(A\) is a \(\mathbb{Q}\)-algebra. In particular, \[ CC_n(f_1,\ldots, f_{n+1})=\exp\mathrm{res}\left(\log(f_1)\frac{df_2}{f_2}\wedge \cdots \wedge \frac{df_{n+1}}{f_{n+1}}\right) \] when \(f_1\) satisfies additional conditions to ensure that \(\log(f_1)\) is well-defined.
In the final section of the article, the authors use the symbol \(CC_n\) to give an explicit description of A. N. Parshin’s local reciprocity map in local class field theory for \(n\)-dimensional local fields of characteristic \(p>0\) [Proc. Steklov Inst. Math. 165, 157–185 (1985; Zbl 0579.12012)]. In particular, they use it to define a higher-dimensional Witt pairing.
The key result in the paper is a uniqueness theorem, characterising the symbol \(CC_n\): Given a functor \(F\) from rings to abelian groups let \(LF\) be the functor \((LF)(A):= F(A((t)))\). Thus \(L^n\mathbb{G}_m(A)=A((t_1))\cdots ((t_n))^*=L_n(A)^*\). By construction, the symbol \(CC_n\) factors as a map of functors \[ (L^n\mathbb{G}_m)^{\times n+1} \to L^nK_{n+1}^M\to \mathbb{G}_m \] where, for a ring \(B\), \(K_n^M(B)\) is the quotient of \((B^*)^{\otimes n}\) by the subgroup generated by the elements of the form \(f_1\otimes \cdots \otimes f_n\) where \(f_{i+1}=1-f_i\) for some \(1\leq i\leq n-1\). Here the second map is a morphism of group-valued functors; in particular, the authors prove in the article that the \(K\)-theory boundary maps \(\partial_m\) are maps of functors. The authors prove (Theorem 8.10) that the group \(\mathrm{Hom}^{\mathrm{gr}}(L^nK^M_{n+1},\mathbb{G}_m)\) of maps of group-valued functors is infinite cyclic with generator \(CC_n\).
The authors’ method of proof of their main theorem is of interest in its own right. The iterated loop groups \(L^n\mathbb{G}_m\), and certain special subfunctors, are naturally represented by inductive systems of affine schemes, or ind-affine schemes. The authors introduce the class of thick ind-cones and develop their properties in order to prove the main theorems.

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