On higher class groups of orders
The authors ``use arithmetic squares and higher-dimensional analogues of the Moore sequence to study the odd-dimensional class groups \({\mathcal C}\ell_{2n- 1}(\Lambda)\), \(n\geq 1\), of an \({\mathcal O}_F\)-order \(\Lambda\) in a semi-simple algebra over a number field \(F\) with ring of integers \({\mathcal O}_F\)''. They prove that ``the only \(p\)-torsion possible in \({\mathcal C}\ell_{2n-1}(\Lambda)\) is for those rational primes \(p\) which lie under a prime of \({\mathcal O}_F\) at which \(\Lambda\) is not maximal'' (and they note \(q\in P_\Lambda\) for rational primes \(q\) lying under primes of \({\mathcal O}_F\) at which \(\Lambda\) is not maximal). For group rings (i.e., \(\Lambda={\mathcal O}_F[G]\) with finite group \(G\)) the only torsion possible is for primes dividing the order of the group. For the general case they assume that for \(q\in P_\Lambda\) the skewfields \(D_i\) appearing in the simple factors of the semi-simple algebra \(A= \prod^r_{i= 1}{\mathcal M}_{n_i}(D_i)\) are fields \(D_i=E_i\), and that the local Quillen-Lichtenbaum conjecture holds for the local fields that appear. Under these assumptions they obtain a surjection \[ \prod^r_{i=1}H^0(E_i, \mathbb{Q}_q/\mathbb{Z}_q(n))\to{\mathcal C}\ell_{2n-1}(\Lambda)(q) \] for \(q\in P_\Lambda\) and \(q\neq 2\). As an application they study the class groups \({\mathcal C}\ell_{4n+1}(\mathbb{Z}[S_r])\) of the symmetric groups \(S_r\) on \(r\) letters. And in the final section they give a description of the product \(\prod^r_{i= 1}H^0(E_i,\mathbb{Q}_q/\mathbb{Z}_q(n))\) for \(\Lambda= \mathbb{Z}[G]\) in the style of Fröhlich.
scientific article; zbMATH DE number 1186178