Vol. 103, No. 1, 1982

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A bigger Brauer group

Joseph L. Taylor

Vol. 103 (1982), No. 1, 163–203

Let R be a commutative ring with identity. In this paper we propose an extension of the concept of central separable R-algebra and use this to define an associated Brauer group for R which contains the classical Brauer group as a subgroup. The essential difference between our notion of central separable algebra and the classical one is that we do not require that the algebra have an identity. As a consequence, our algebras need not be finitely generated or projective as R-modules. Nevertheless, with equality defined using an appropriate version of Morita equivalence and tensor product providing the operation, we obtain a tractable extension of the Brauer group. If R is a Henselian local ring with algebraically closed residual field, our Brauer group is trivial. If R is the algebra of complex valued continuous functions on a compact Hausdorff space our Brauer group is the full integral third Čech cohomology group of the underlying space, while the classical Brauer group is just the torsion subgroup.

Mathematical Subject Classification 2000
Primary: 13A20
Secondary: 46J05
Received: 8 December 1980
Published: 1 November 1982
Joseph L. Taylor