On the Teichmüller cocycle map
Let G be a finite group acting as automorphisms of a commutative ring S, and let R be the subring of elements fixed by G. L. N. Childs defined a group \(H^ 3_{0,1}(G)\) and investigated a sequence of groups: \[ 1\to Br(S/R)\to Br(R)\to H^ 0(G,Br(S))\to H^ 3_{0,1}(G), \] where Br(S) denotes the Brauer group of Azumaya S-algebras and Br(S/R) is the kernel of the group homomorphism of Br(R) into Br(S) induced by extension of the ring of scalars [see \textit{L. N. Childs}, J. Algebra 23, 1-17 (1972; Zbl 0243.16003)]. If S is a Galois extension of R, then the sequence is exact, \(H^ 0(G,Br(S))\) is isomorphic to the Amitsur cohomology group \(H^ 0(S/R,Br)\), and \(H^ 3_{0,1}(G)\) is isomorphic to the group \(H^ 3(J)\) defined by \textit{O. E. Villamayor} and \textit{D. Zelinsky} [J. Pure Appl. Algebra 10, 19-55 (1977; Zbl 0383.13002)]. The author gives another description of \(H^ 3_{0,1}(G)\) and an explicit definition of a homomorphism t of \(H^ 0(G,Br(S))\) into \(H^ 3_{0,1}(G)\). When S is a Galois extension of R, the above sequence is exact for this \(\hom omorphism\quad t;\) and the author conjectures that t coincides with the homomorphism of \(H^ 0(S/R,Br)\) into \(H^ 3(J)\) described by Villamayor and Zelinsky.
scientific article; zbMATH DE number 4136084
scientific article