Trace forms of \(G\)-Galois algebras in virtual cohomological dimension 1 and 2
Let \(k\) be a field of characteristic different from~\(2\) and let \(G\) be a finite group. Let \(\Gamma_k\) denote the absolute Galois group of \(k\). A \(G\)-Galois algebra over \(k\) is a finite etale (commutative) \(k\)-algebra that is Galois over \(k\) with group \(G\). The trace form of a \(G\)-Galois algebra \(L\) is the quadratic form \(q_L\) on \(L\) defined by \(q_L(x)=\text{Tr}_{L/k}(x^2)\). It is a \(G\)-form in the sense that it is invariant under the action of \(G\) on \(L\). On the other hand, to each \(G\)-Galois algebra \(L\) over \(k\), Bayer-Fluckiger and Serre attached a continuous homomorphism \(\varphi_L:\Gamma_k\to G\) reflecting the action of \(\Gamma_k\) on the set of \(k\)-algebra homomorphisms of \(L\) in a separable closure of \(k\), see [Am. J. Math. 116, No. 1, 1--64 (1994; Zbl 0804.12004)]. The reduction modulo the subgroup \(G^2\subset G\) generated by the squares is a continuous homomorphism \(\bar\varphi_L\in\text{Hom}_{\text{{cont}}}(\Gamma_k,G/G^2)\). In the paper under review, the authors show that the assignment \(L\mapsto \bar\varphi_L\) induces a bijection between the set of isomorphism classes of trace forms of \(G\)-Galois algebras and \(\text{Hom}_{\text{{cont}}}(\Gamma_k,G/G^2)\) when the \(2\)-cohomological dimension of \(k\) is at most~\(1\). This result is shown to extend to the case where the virtual \(2\)-cohomological dimension of \(k\), i.e., the \(2\)-cohomological dimension of \(k(\sqrt{-1})\), is at most \(1\), provided that a suitable group taking into account the orderings of \(k\) is substituted for \(\text{Hom}_{\text{{cont}}}(\Gamma_k,G/G^2)\). In the case of virtual \(2\)-cohomological dimension \(2\), the authors show that for \(G\)-Galois algebras \(L\), \(L'\) over \(k\), the ``double'' \(G\)-forms \(q_L\oplus q_L\) and \(q_{L'}\oplus q_{L'}\) are isomorphic if and only if the following two conditions hold: (i) the \(G\)-forms \(q_L\) and \(q_{L'}\) are isomorphic over all real closures of \(k\), and (ii) the cup products \(\varphi_L\cup(-1)\) and \(\varphi_{L'}\cup(-1)\) are equal in \(H^2(\Gamma_k,G/G^2)\). This was shown in [Algebra Anal. 11, No. 3, 1--19 (1999; Zbl 0957.12004)] under the additional hypothesis that \(k\) has the strong approximation property.
scientific article; zbMATH DE number 2164605