Operations on locally free classgroups (Q1298158)

For every number field \(K\) and every finite group \(\Gamma\), the so-called locally free class group Cl\((O_K\Gamma)\) (which is just the Picard group if \(\Gamma\) is abelian) is given, via Fröhlich's Hom description, as a suitable factor group of Hom\((R,I)\), where \(R\) is the representation ring of \(\Gamma\) over \(\bar K\) and \(I\) the adele ring of \(\bar K\). From this description and the usual Adams operations in representation theory, Cassou-Noguès and Taylor have constructed, under suitable hypotheses, Adams operations \(\psi_k^{CNT}\) on Cl\((O_K\Gamma)\). The purpose of the article under review is, foremost, to find a more direct interpretation of \(\psi_k^{CNT}\) in terms of locally free modules. In this, the article succeeds very well: if gcd\((k,|\Gamma|)=1\), the map \(\psi_k^{CNT}\) is identified with a symmetric power operation on the K-group K\(_0(O_K\Gamma)\). Interestingly, it is not the \(k\)th but the \(k'\)th symmetric power operation \(\sigma_{k'}\) one has to take here, where \(k'\) is an inverse of \(k\) modulo \(|\Gamma|\). There are also algebraic Adams operations \(\psi\) on K\(_0(O_K\Gamma)\), and it is proved as well that \(k\cdot \psi_k^{CNT} = \psi_{k'}\) under the same condition. If \(\Gamma\) is the Galois group of a tame Galois extension \(N/K\), Burns and Chinburg previously found another description of \(\psi_k^{CNT}\) via modules, more precisely: they expressed \(\psi_k^{CNT}(A)\) for any \(\Gamma\)-stable ideal in \(N\) in terms of powers of the different of \(N/K\) and of powers of \(A\). On comparison with the description given in the present paper, it then turns out that (I quote) ``more or less, the formula of Burns and Chinburg is a strengthening of the equivariant Adams-Riemann-Roch formula.'' This paper provides symmetric power, exterior power, and Adams operations not only for locally free class groups but also for all higher K-groups of (twisted) group rings. The starting point (Prop. 1.1) is a very explicit result on projective modules, but later on a fair amount of general technique is needed. One important point is the following: The three operations commute with the connecting homomorphism from K\(_1(K_p\Gamma)\) to K\(_0(O_K\Gamma)\). This fact is proved twice: by algebraic computation, and by a more conceptual argument using topological machinery.