Higher-dimensional Contou-Carrère symbol and continuous automorphisms (Q524115)

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 \(A\).