Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
38(1), 33 - 72 (1997)

Discriminant Algebras and Adjoints of Quadratic Forms

Ottmar Loos

Institut für Mathematik, Universität Innsbruck A-6020 Innsbruck, Austria

Abstract: The discriminant algebra of a nonsingular quadratic form $q$ over a commutative ring is usually defined as the centralizer of the even Clifford algebra in the full Clifford algebra. U. Haag [Diskriminantenalgebren quadratischer Formen, Arch. Math. 57 (1991), 546-554] introduced the discriminant algebra of a possibly singular quadratic form on a finitely generated free module by working with generic forms and then applying specialization. His method depends on a choice of basis, and although the result is up to isomorphism independent of that choice, an extra effort is required to extend the definition to non-free finitely generated projective modules. In this paper, we present a definition of the discriminant algebra in a natural basis-free way which avoids the Clifford algebra entirely. The construction involves an Amitsur 1-cocycle on bilinear forms representing $q$. As a byproduct, we obtain an explicit formula for the Dickson homomorphism. The functor associating with a quadratic module its discriminant algebra is multiplicative with respect to orthogonal sums, and a suitably defined product of quadratic algebras, which generalizes the well-known product in the separable case. We also show that the discriminant algebra admits a natural homomorphism into the centralizer of the even Clifford algebra in the full Clifford algebra which is an isomorphism in the nonsingular or semiregular case. This makes use of Bourbaki's $\lambda$-operations between Clifford algebras, for which we derive an explicit formula. For a quadratic form $q$ of odd rank $2n+1$, we introduce a quadratic form $q^\sharp$ with base point of rank 2n+2, called the {\it adjoint form}, which is constructed from the second-to-the-highest exterior power of the base module by the same method as the discriminant algebra in the even rank case. Similarly to the discriminant algebra, there is a natural homomorphism into the Clifford algebra, and its image is contained in a Jordan subalgebra $J$ which we call the {\it Jordan center \/} since it is defined by suitable centralizer conditions. In case $q$ is nonsingular of even rank or semiregular of odd rank, the structure of $J$ is completely determined.

Keywords: Quadratic form, discriminant algebra

Classification (MSC91): 11Exx, 15A63

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