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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 $$A$$.

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