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

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.