The Brauer-Wall group of a commutative ring

Small, Charles

doi:10.1090/S0002-9947-1971-0276218-4

The Brauer-Wall group of a commutative ring
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Trans. Amer. Math. Soc. 156 (1971), 455-491 Request permission

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.

References

Maurice Auslander and Oscar Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367–409. MR 121392, DOI 10.1090/S0002-9947-1960-0121392-6
Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491

Topics in algebraic K-theory

S. U. Chase, D. K. Harrison, and Alex Rosenberg, Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. 52 (1965), 15–33. MR 195922
G. J. Janusz, Separable algebras over commutative rings, Trans. Amer. Math. Soc. 122 (1966), 461–479. MR 210699, DOI 10.1090/S0002-9947-1966-0210699-5

Fondements de la K-theorie

Applications algébriques de la cohomologie des groupes

Théorie des algèbres simples

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Jean-Pierre Serre, Corps locaux, Publications de l’Institut de Mathématique de l’Université de Nancago, VIII, Hermann, Paris, 1962 (French). Actualités Sci. Indust., No. 1296. MR 0150130
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