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Generalized crossed products applied to maximal orders, Brauer groups and related exact sequences. (English) Zbl 0538.16003
The results in this paper circle around the following main idea: if A is a certain maximal order over a commutative ring C and A contains a commutative extension S of C such that G acts as a group of C- automorphisms of S such that \(S^ G\) (the fixed ring of G) equals C, then A is a generalized crossed product over S with respect to G, i.e. \(A=\oplus_{\sigma \in G}S_{\sigma},\quad S_ e=S\quad and\quad S_{\sigma}S_{\tau}=S_{\sigma \tau}\) for all \(\sigma,\tau \in G.\) The authors actually consider the following situations: A is a maximal Krull order over a Dedekind domain; A is a relative Azuyama algebra, or in particular a reflexive Azumaya algebra in the sense of Yuan; A is a common Azumaya algebra. Certain extra conditions have to be imposed on S and these have the effect that S becomes a ”relative” or a ”weak” Galois extension of C.
Reviewer: C.Năstăsescu

MSC:
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16W20 Automorphisms and endomorphisms
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