Graded Brauer groups. (English) Zbl 0125.01904

Wall, C. T. C.

J. Reine Angew. Math. 213, 187-199 (1964).

Author’s introduction: This paper arose out of the observation that the definition of a Clifford algebra as an invariant of a quadratic form is made awkward by the fact that the algebra corresponding to the orthogonal direct sum of two quadratic forms is not simply the tensor product of their separate Clifford algebras. In fact, as is well known, the Clifford algebra admits a grading modulo 2, and we must consider the graded tensor product.
The object of this paper is to perform the theory of the Brauer group for graded algebras. We first define the class of central simple graded algebras over a field \(A\), and then proceed to investigate their structure: we obtain a complete description. We then show that the class is closed under graded tensor products, and that the algebras, taken with a suitable equivalence relation, define a group, which we christen the graded Brauer group of \(k\). The structure of this group is determined in terms of that of the ordinary Brauer group of \(A\); in particular, if \(k\) is the real field, our group is cyclic of order \(8\). We then observe that taking the Clifford algebra of a quadratic form over \(k\) defines a homomorphism of the Witt group of \(k\) to the graded Brauer group; this image tells us essentially the determinant of the form, and an ungraded central simple algebra (one of the two Clifford algebras). A last paragraph clears up the case when \(k\) has characteristic \(2\), when a number of results are somewhat different; we obtain an equally complete theory, and, for example, the invariants of a quadratic form now reduce to the (ungraded) Clifford algebra and the Arf invariant.

Reviewer: T. A. Springer

MSC:

16K50	Brauer groups (algebraic aspects)
15A66	Clifford algebras, spinors

Keywords:

graded Brauer groups