Hinda Hamraoui: Un facteur direct canonique de la $K$-th\'eorie d'anneaux d'entiers alg\'ebriques non exceptionnels

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Submission: 2000, Sep. 10

Let $R$ be a ring of algebraic integers, non-exceptional in the sense of \cite{HS} and not containing $\sqrt{-1}$, and let $K$ be its quotient field. Let $\xi_{2^m}$ be a $2$-primary root of unity of maximal order $2^m$ in $K(\sqrt{-1})$ and $\mu_{2^m }$ the cyclic group generated by $\xi_{2^m}$. In \cite{HS}, Harris and Segal have given residue class fields $k$ of $R$ such that $\mu_{2^{\infty}}(R(\sqrt{-1})) \to \mu_{2^{\infty}}(k (\sqrt{-1}))$ is an isomorphism, equivariant for the Galois action of $\Z/2$. In particular, the semi-dihedral group $\Delta:\Z/2\ltimes \mu_{2^m }$ is a common subgroup to $\GL(2,k)$ and $\GL(2,R)$. The natural embeddings $\Delta\to \GL(2,k)$ and $\Delta\to \GL(2,R)$ induce maps $\phi_k:\B(\Sigma_\infty\int \Delta)^+\rightarrow \B\GL(k)^+$, $\phi_R:\B(\Sigma_\infty\int \Delta)^+\rightarrow \B\G L(R)^+$ and corresponding maps $L\phi_k$, $L\phi_R$ after localisation at $2$. Let $s$ be any right inverse of $L\phi_k$ see \cite{HS}. The aim of this Note is to extend the results of \cite[theorem 4.1 (2)]{DFM} to the case where $R$ is non exceptional and $\sqrt{-1}$ does not belong to $R$, i.e. to prove that the triangle \eqref{3} above is homotopy commutative.

2000 Mathematics Subject Classification: 19F99, 11R70

Keywords and Phrases: K-theory, algebraic integers

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