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Transactions of the American Mathematical Society

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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.

References [Enhancements On Off] (What's this?)

  • [1] M. Auslander and O. Goldman, The Brauer group of a commutative ring, Trans. Amer. Math. Soc. 97 (1960), 367-409. MR 22 #12130. MR 0121392 (22:12130)
  • [2] H. Bass, Algebraic K-theory, Benjamin, New York, 1968. MR 40 #2736. MR 0249491 (40:2736)
  • [3] -, Topics in algebraic K-theory, Lecture Notes, Tata Institute, Bombay, 1967.
  • [4] S. U. Chase, D. K. Harrison and A. Rosenberg, Galois theory and Galois cohomology of commutative rings, Mem. Amer. Math. Soc. No. 52 (1965). MR 33 #4118. MR 0195922 (33:4118)
  • [5] G. Janusz, Separable algebras over commutative rings, Thesis, University of Oregon, Eugene, Ore., 1965. MR 0210699 (35:1585)
  • [6] M. Karoubi, Fondements de la K-theorie, Facultédes Sciences de l'Université d'Alger, 1966/67.
  • [7] 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.
  • [8] -, Corps locaux, Actualités Sci. Indust., no. 1296, Hermann, Paris, 1962. MR 27 #133. MR 0150130 (27:133)
  • [9] C. T. C. Wall, Graded Brauer groups, J. Reine Angew. Math. 213 (1963/64), 187-199. MR 29 #4771. MR 0167498 (29:4771)

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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

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