Twists of matrix algebras and Brauer groups
Let \(F\) be a global field and \(G_F\) the absolute Galois group, \(G_F = \text{Gal} (\overline F/F)\), of \(F\). For a finite dimensional \(F\)- algebra \(A\), the nonabelian cohomology set \(H^1 (G_F, \Aut_{\overline F} (A \otimes \overline F))\) can be thought of as the set \(S\) of \(F\)-isomorphism classes of \(F\)-algebras \(B\) such that \(B \otimes_F \overline F\) is \(\overline F\)-isomorphic to \(A \otimes \overline F\). Namely, given a 1-cocycle \(u\) on \(G_F\), then \(u\) defines a \(G_F\) action on \(A \otimes \overline F\) by \(s \cdot a = u_s ((1 \otimes s) (a))\), where \(a \in A \otimes \overline F\), \(s\in G_F\). Then the twist algebra \(A(u) = (A \otimes \overline F)^{G_F}\). Let \(\alpha\) be a continuous character of \(G_F\) into \(\overline F^*/F^*\). Let \(E\) be a finite extension of \(F\) in \(\overline F\) such that the multiplicative group \(E^*/F^*\) contains all values of \(\alpha\). \(E^*/F^*\) can be embedded as a subgroup of the group automorphisms \(\Aut_F (\text{End}_FE)\) \((e \in E^*\) acting as inner automorphism on the elements of \(\text{End}_F E)\). So \(\alpha\) is a continuous group morphism, and therefore it can be regarded as a 1- cocycle. Let \([\alpha]\) be the corresponding cohomology class of the twist algebra \((\text{End}_FE) (\alpha)\) in \(Br(F) = H^2 (G_F, \overline F^*)\). In the paper it is shown that the correspondence \(\alpha \mapsto [\alpha]\) defines a surjective group morphism; \(t_F : \Hom_{ct} (G_F, \overline F^*/F^*) \to Br(F)\), in the case \(F\) is a number field. In the function field case, \(t_F\) is surjective on the non \(p\)- part, where \(p\) is the characteristic of \(F\). Moreover if \(K\) is a global subfield of \(F\) then the same holds for the map \(s_F = t_F \circ res\), although it is known by class field theory that \(res\) is not surjective.
scientific article; zbMATH DE number 741480