Cohomologie des groupes et corps d'invariants multiplicatifs. (Cohomology of groups and fields of multiplicative invariants) (Q1121947)

Let \(\Gamma\) be a finite group acting linearly and faithfully on a complex finite dimensional vector space V. The following problem is well known: is the field of invariants \({\mathbb{C}}(V)^{\Gamma}\) pure transcendental over \({\mathbb{C}}?\) The unramified Brauer group \(Br^{nr}({\mathbb{C}}(V)^{\Gamma})\) is very useful for its investigation. It is known that \(Br^{nr}({\mathbb{C}}(V)^{\Gamma})=0\) if the extension \({\mathbb{C}}(V)^{\Gamma}/{\mathbb{C}}\) is pure transcendental. Saltman constructed a finite group acting faithfully on V such that \(Br^{nr}({\mathbb{C}}(V)^{\Gamma})\neq 0\). - Then Bogomolov and Saltman proved independently the following theorem 1: \[ Br^{nr}({\mathbb{C}}(V)^{\Gamma})=Ker[H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}})\to \prod_{A}H^ 2(A,{\mathbb{Q}}/{\mathbb{Z}}) ], \] where A runs over all abelian subgroups of \(\Gamma\). Let G be a finite group and M be a free \({\mathbb{Z}}\)-module of finite type such that G acts on M faithfully. Let \({\mathbb{C}}(M)\) be the field of fractions of the group ring of the abelian group M with coefficients in \({\mathbb{C}}\). Saltman proved the following theorem 2: \[ Br^{nr}({\mathbb{C}}(M)^ G)=Ker([H^ 2(G,{\mathbb{Q}}/{\mathbb{Z}}\oplus M)\to \prod_{B}H^ 2(B,{\mathbb{Q}}/{\mathbb{Z}}\oplus M) ]), \] where B runs over subgroups of G which are isomorphic to quotients of \({\mathbb{Z}}^ 2.\) The aim of the paper under review is to give another proof of theorem 2 by using cohomological methods.