Semisimple algebras, Galois actions and group cohomology
Let \(G\) be a finite group acting on a commutative field \(K\) via \(\tau : G \to \text{Aut}(K)\). The article under review investigates when crossed products \(R = K^ \alpha_ \tau G\) for \(\alpha \in H^ 2(G, K^*)\) are semisimple. Since this is known to be always the case for \(\text{char }K = 0\), let \(\text{char }K = p > 0\). In this case, the structure of \(H = \text{Ker }\tau\) and, more precisely, of \(\overline{H} = H/O_{p'} (H)\) turns out to be decisive. The following criterion is quickly established in the article, using a reduction to the case of a trivial action \(\tau\), due to Aljadeff and Rosset, and a Sylow reduction due to Passman: \(R\) is semisimple if and only if (1) \(\overline{H}\) is abelian; so \(H =O_{p'} (H) \rtimes P\) with \(P \cong\times^ r_ 1 \mathbb{Z}/p^{e_ i} \mathbb{Z}\), and (2) \(\text{res}^ G_ p (\alpha) \in H^ 2 (P, K^*) \cong \otimes^ r_ 1 K^*/(K^*)^{p^{e_ i}}\) is represented by an \(r\)-tuple (\(a_ 1 (K^*)^{p^{e_ 1}}, \dots, a_ r(K^*)^{p^{e_ r}}\)) such that \(\{a_ 1, \dots, a_ r\}\) is \(p\)- independent over \(K^ p\). -- The authors then turn to the question whether, given \(G\), \(K\), and \(\tau\), there exists \(\alpha \in H^ 2 (G,K^*)\) to that \(R\) is semisimple. In view of the above criterion, it is necessary that \(\overline{H}\) be an abelian \(p\)-group of rank not exceeding the \(p\)-degree of \(K\) over \(K^ p\). The second main result of the article shows that if, in addition, \(\overline {H}\) has an ascending series of \(G\)-invariant subgroups with cyclic quotients then \(\alpha\) can indeed be found. The latter condition is automatic for \(G\) supersolvable, and it is a consequence of the above necessary condition if \(\tau\) is trivial or \(K\) is a local field. The question as to whether the above necessary condition is always sufficient remains open in general.
ascending series of \(G\)-invariant subgroups
scientific article; zbMATH DE number 609787