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).


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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