## 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 C_{k}(ϕ) 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→H2(GKϕ∗)→Cliffk(ϕ)→Ck(ϕ)→H3(GKϕ∗). 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.

Original language | English |
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Pages (from-to) | 955-985 |

Number of pages | 31 |

Journal | Israel Journal of Mathematics |

Volume | 247 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2022 |

### Bibliographical note

Publisher Copyright:© 2022, The Hebrew University of Jerusalem.

## ASJC Scopus subject areas

- General Mathematics