Realization-obstruction exact sequences for Clifford system extensions

Abstract

For every action φ ∈ Hom(G, Aut_k(K)) of a group G on a commutative ring K we introduce two abelian monoids. The monoid Cliff_k(φ) consists of equivalence classes of strongly G-graded algebras of type φ up to G-graded Clifford system extensions of K-central algebras. The monoid ${{\cal C}_k}(\phi )$ consists of equivariance classes of homomorphisms of type φ from G to the Picard groups of K-central algebras (generalized collective characters). Furthermore, for every such φ there is an exact sequence of abelian monoids

$$0 \to {H^2}(G,K_\phi ^ \ast ) \to {\rm{Clif}}{{\rm{f}}_k}(\phi ) \to {{\cal C}_k}(\phi ) \to {H^3}(G,K_\phi ^ \ast ).$$

This sequence describes the obstruction to realizing a generalized collective character of type φ, that is it determines if such a character is associated to some strongly G-graded k-algebra. The rightmost homomorphism is often surjective, terminating the above sequence. When φ is a Galois action, then the well-known restriction-obstruction sequence of Brauer groups is an image of an exact sequence of sub-monoids appearing in the above sequence.