## The Brauer-Wall group of a commutative ring

HTML articles powered by AMS MathViewer

- by Charles Small PDF
- 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**
—, - 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
M. Karoubi, - 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** - C. T. C. Wall,
*Graded Brauer groups*, J. Reine Angew. Math.**213**(1963/64), 187–199. MR**167498**, DOI 10.1515/crll.1964.213.187

*Topics in algebraic K-theory*, Lecture Notes, Tata Institute, Bombay, 1967.

*Fondements de la K-theorie*, Facultédes Sciences de l’Université d’Alger, 1966/67. J.-P. Sérre,

*Applications algébriques de la cohomologie des groupes*:

*Théorie des algèbres simples*, Séminaire H. Cartan 1950/51, exposés 5-7, Secrétariat mathématique, Paris, 1955. MR

**17**, 1117.

## Additional Information

- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**156**(1971), 455-491 - MSC: Primary 13.90
- DOI: https://doi.org/10.1090/S0002-9947-1971-0276218-4
- MathSciNet review: 0276218