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Reduced norms in the K-theory of orders. (English) Zbl 0358.16021
Let $$A$$ be an $$R$$-order in a semisimple algebra $$B$$. In this paper general formulae of an idelic nature are obtained for the subgroups $$Cl(A)$$ and $$G^{f_0^l}(A)$$ of $$K_0(A)$$ and $$G_0(A)$$ consisting of those elements killed by every map got from localization at the maximal ideals of $$R$$. These formulae are similar to, and include, that published by several authors for $$Cl(A)$$ in the arithmetic case. The kernel $$D(A)$$ of the extension map from $$Cl(A)$$ to $$Cl(M)$$, where $$M$$ is a maximal order containing $$A$$, is also considered. It is rapidly deduced that if $$B$$ is simple (that $$A$$ contains all central idempotents is enough) then $$D(A)$$ is the kernel of the Cartan map from $$Cl(A)$$ to $$G^{f_0^l}(A)$$ and that if $$A$$ is hereditary then $$D(A)$$ is $$0$$. The cokernel of the Cartan map is also considered. The author concludes with a demonstration of the idelic approach to class groups. After proving some results about reduced norms he shows that the 2-part of the class group of the quaternion group of order $$4^{\ell r}$$ ($$\ell$$ prime) is elementary abelian of order $$2^r$$ (he relies on a result of Fröhlich that the class group has a quotient of this type).

MSC:
 16E20 Grothendieck groups, $$K$$-theory, etc. 16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) 16Kxx Division rings and semisimple Artin rings 11R37 Class field theory 11R52 Quaternion and other division algebras: arithmetic, zeta functions
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References:
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