A COHOMOLOGICAL APPROACH TO THE BRAUER-LONG GROUP AND THE GROUPS OF GALOIS EXTENSIONS AND STRONGLY GRADED RINGS

Let G be a finite abelian group, and R a commutative ring. The Brauer-Long group BD(R, G) is described by an exact sequence [GRAPHICS] where BD(s) (R, G) is a product of etale cohomology groups, and Im beta is a kind of orthogonal subgroup of Aut(G x G*)(R). This sequence generalizes some other well-known exact sequences, and restricts to two split exact sequences describing Galois extensions and strongly graded rings.