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Let G be a finite abelian group, and R a commutative ring.The Brauer-Long group BD(.R, G) is described by an exact sequencewhere BDS (R,G) is a product of étale cohomology groups, and Im/J 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. IntroductionIn 1964, Wall introduced a Brauer group of classes of Z/2Z-graded algebras over a field with multiplication induced by a twisted tensor product, to study the Witt ring of quadratic forms (cf.[37]).This group was named the Brauer-Wall group, and, since then, some various generalizations to commutative rings and to finite abelian groups have been introduced by Bass and Small [2, 31], Knus [20], and Childs, Garfinkel and Orzech [13].In [23], Long introduced a Brauer group consisting of equivalence classes of G-dimodule algebras over a commutative ring R, and for an arbitrary abelian group G.This new invariant of R and G contained all the previous generalizations of the Brauer-Wall group as subgroups, and, moreover, it contains other invariants of R, namely the Brauer group of R, the Galois extensions of R with group G, the strongly G-graded rings having R as part of degree zero, and the automorphism group of G. Afterwards, this new Brauer group was called the Brauer-Long group, denoted by BD(Pv, G).In the literature, many calculations have been done in some special cases; some authors restricted attention to some subgroups of the Brauer-Long group [3,4,6,14,28], while others considered only the case where G is cyclic [7,16] or a product of cyclic groups of the same order [12].In most

A Cohomological Approach to the Brauer-Long Group and the Groups of Galois Extensions and Strongly Graded Rings

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