Realization-obstruction exact sequences for Clifford system extensions

Abstract

For every action $\phi\in\text{Hom}(G,\text{Aut}_k(K))$ of a group $G$ on a commutative ring $K$ we introduce two abelian monoids. The monoid $\text{Cliff}_k(\phi)$ consists of equivalent classes of $G$-graded Clifford system extensions of type $\phi$ of $K$-central algebras. The monoid $\mathcal{C}_k{(\phi)}$ consists of equivariant classes of generalized collective characters of type $\phi$ from $G$ to the Picard groups of $K$-central algebras. Furthermore, for every such $\phi$ there is an exact sequence of abelian monoids $$0\to H^2(G,K^*_{\phi})\to\text{Cliff}_k(\phi)\to\mathcal{C}_k{(\phi)}\to H^3(G,K^*_{\phi}).$$ The rightmost homomorphism is often surjective, terminating the above sequence. When $\phi$ is a Galois action, then the restriction-obstruction sequence of Brauer groups is an image of an exact sequence of sub-monoids of this sequence.