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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Brauer-Wall group of a commutative ring


Author: Charles Small
Journal: Trans. Amer. Math. Soc. 156 (1971), 455-491
MSC: Primary 13.90
MathSciNet review: 0276218
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Abstract: Let k be a commutative ring (with 1). We work with k-algebras with a grading $ \bmod\;2$, and with graded modules over such algebras. Using graded notions of tensor product, commutativity, and morphisms, we construct an abelian group $ {\rm {BW}}(k)$ whose elements are suitable equivalence classes of Azumaya k-algebras. The consruction generalizes, and is patterned on, the definition of the Brauer group $ {\rm {Br}}(k)$ given by Auslander and Goldman. $ {\rm {Br}}(k)$ is in fact a subgroup of $ {\rm {BW}}(k)$, and we describe the quotient as a group of graded quadratic extensions of k.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0276218-4
PII: S 0002-9947(1971)0276218-4
Keywords: Brauer group of a commutative ring, separable algebra, Azumaya algebra, graded algebra, Galois extension of commutative rings, quadratic extension of commutative rings
Article copyright: © Copyright 1971 American Mathematical Society