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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A cohomological approach to the Brauer-Long group and the groups of Galois extensions and strongly graded rings


Authors: S. Caenepeel and M. Beattie
Journal: Trans. Amer. Math. Soc. 324 (1991), 747-775
MSC: Primary 16H05; Secondary 13A20
DOI: https://doi.org/10.1090/S0002-9947-1991-0987160-8
MathSciNet review: 987160
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Abstract: Let $G$ be a finite abelian group, and $R$ a commutative ring. The Brauer-Long group $\operatorname {BD} (R,G)$ is described by an exact sequence \[ 1 \to {\operatorname {BD} ^s}(R,G) \to \operatorname {BD} (R,G)\xrightarrow {\beta }\operatorname {Aut} (G \times {G^{\ast }})(R)\] where ${\operatorname {BD} ^s}(R,G)$ is a product of étale cohomology groups, and Im $\beta$ is a kind of orthogonal subgroup of $\operatorname {Aut} (G \times {G^{\ast }})(R)$. This sequence generalizes some other well-known exact sequences, and restricts to two split exact sequences describing Galois extensions and strongly graded rings.


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Keywords: Brauer group, Azumaya algebra, Galois extension, étale cohomology, strongly graded ring, dimodule algebra
Article copyright: © Copyright 1991 American Mathematical Society