Abstract
Based on the construction of the discriminant algebra of an even-ranked quadratic form and Rost’s method of shifting quadratic algebras, we give an explicit rational construction of the discriminant algebra of finite-rank algebras and, more generally, of quadratic trace modules, over arbitrary commutative rings. The discriminant algebra is a tensor functor with values in quadratic algebras, and a symmetric tensor functor with values in quadratic algebras with parity. The automorphism group of a separable quadratic trace module is a smooth, but in general not reductive, group scheme admitting a Dickson type homomorphism into the constant group scheme Z 2.
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Loos, O. Discriminant algebras of finite rank algebras and quadratic trace modules. Math. Z. 257, 467–523 (2007). https://doi.org/10.1007/s00209-007-0123-6
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DOI: https://doi.org/10.1007/s00209-007-0123-6