zbMATH — the first resource for mathematics

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
Full Text: DOI
[1] Bak, A; Scharlau, W, Grothendieck and Witt groups of orders and finite groups, Invent. math., 23, 207-240, (1974) · Zbl 0288.16016
[2] Bass, H, Algebraic K-theory, (1968), Benjamin New York · Zbl 0174.30302
[3] {\scP. Draxl}, SK1 von Algebren fiber vollständig diskret bewerteten Körpern and Galois cohomologie abelscher Köpererweiterung, J. Reine Angew. Math., to appear.
[4] Ferrar, W.L, Higher algebra, (1948), Oxford University Press · Zbl 0032.00101
[5] Fröhlich, A, Locally free modules over arithmetic orders, J. reine angew. math., 274/275, 112-138, (1975) · Zbl 0316.12013
[6] Fröhlich, A, Module invariants and root numbers for quaternion fields of degree 4lr, (), 393-399 · Zbl 0304.12008
[7] Fröhlich, A; Keating, M.E; Wilson, S.M.J, The class groups of quternion and dihedral 2-groups, Mathematika, 21, 64-71, (1974) · Zbl 0303.12006
[8] Galovich, S, The class group of a cyclic p-group, J. algebra, 30, 368-387, (1974) · Zbl 0282.13004
[9] Heller, A; Reiner, I, Grothendieck groups of orders in semi-simple algebras, Trans. amer. math. soc., 112, 344-355, (1964) · Zbl 0127.25803
[10] Jacobinski, H, Two remarks about hereditary orders, (), 1-8 · Zbl 0216.06501
[11] Jančevskii, V.I, The commutator subgroups of simple algebras with surjective reduced norms, Soviet math. dolk., 16, 492-495, (1975) · Zbl 0333.20033
[12] Wall, C.T.C, On the classification of Hermitian forms. IV, adèle rings, Invent. math., 23, 241-260, (1974) · Zbl 0278.16017
[13] Wall, C.T.C, On the classification of Hermitian forms. V, global rings, Invent. math., 23, 261-292, (1974) · Zbl 0278.16018
[14] Wilson, S.M.J, On the K-theory of twisted group rings, Thesis, (1972), London · Zbl 0317.16012
[15] Wilson, S.M.J, Twisted group rings and ramification, (), 311-330, 31 · Zbl 0317.16013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.