Opérations d'Adams et groupe des classes d'algèbre de groupe. (Adams operations and class groups of group algebras) (Q1082372)

The main result of this paper is that for \(\Gamma\) a finite group (of exponent n), Adams operations \(\Psi_ k\) can be defined on the groups \(K_ 0({\mathbb{Z}}\Gamma)\) and \(K_ 0 T({\mathbb{Z}}\Gamma)\), where T(\({\mathbb{Z}}\Gamma)\) is the category of \({\mathbb{Z}}\)-torsion \({\mathbb{Z}}\Gamma\)- modules of finite projective dimension. More generally, if \({\mathfrak O}_ K\) is the ring of integers in the field K, a finite extension of \({\mathbb{Q}}\) or of some \(\ell\)-adic field, with algebraic closure \(\bar K\) and Galois group \(\Omega_ K\), one has an isomorphism of \(K_ 1(K\Gamma)\) on \(Hom_{\Omega_ K}(G_ 0(\bar K\Gamma),\bar K^*)\) where the Grothendieck group \(G_ 0(\bar K\Gamma)\) admits Adams operations coming from exterior powers of \(\bar K\Gamma\)-modules. If K is local, \(K_ 0 T({\mathfrak O}_ K\Gamma)\) is a quotient of \(K_ 1(K\Gamma)\), and the burden of the proof consists in showing that the operations pass to the quotient. This is easy for \(\ell\) prime to \(| \Gamma |\); otherwise the proof consists in reducing first (by induction theorems) to the \({\mathbb{Q}}\)-\(\ell\)-elementary case; then (using certain projection maps) to the case of \(\ell\)-groups, where Taylor's \(\ell\)-adic logarithms are available. From this main local result the authors deduce, for example, Adams operations in \(K_ 0 T({\mathfrak O}_ K\Gamma)\) for K a number field with discriminant prime to n and operations \(\Psi_ k\) (k odd) in the class groups CL(\({\mathfrak O}_ K\Gamma)\). The authors show that these operations anticommute in a certain sense with the Cartan homomorphism. Applications are given to prove the vanishing of certain Galois cohomology groups of \(Wh'({\mathbb{Z}}_{\ell}\Gamma)\) and to begin the investigation of Stickelberger-type relations annihilating the class groups.