The crossed product theorem for projective Schur algebras (Q2710438)

The authors investigate relations between cohomology groups and some Azumaya algebras and prove the following main results: (1) Let \(L\) be a finite Galois radical extension of a field \(k\). Then a radical \(k\)-algebra split by \(L\) corresponds to a cohomology class in \(H^2(L/k)\) that is an image of some elements in \(H^2(L/k,\Omega)\), for some multiplicative subgroup \(\Omega<L^*\) such that \(L=k(\Omega)\) and \(\Omega k^*/k^*\) is torsion. The converse is also true; (2) Let \(L\) be a finite Galois radical extension of a field \(k\) and let \(H^2_0(L/k)\) be the image of the homomorphism \(H^2(L/k,\Omega)\to H^2(L/k)\), for some multiplicative subgroup \(\Omega\) of \(L^*\) with \(L=k(\Omega)\) and \(\Omega k^*/k^*\) torsion. The radical subgroup \(R(L/k)\) of the Brauer group \(B(k)\) is isomorphic to the subgroup \(H^2_0(L/k)\) of \(H^2(L/k)\).