A New Discriminant Algebra Construction

A discriminant algebra operation sends a commutative ring $R$ and an $R$-algebra
$A$ of rank $n$ to an $R$-algebra $\Delta_{A/R}$ of rank 2 with the same
discriminant bilinear form. Constructions of discriminant algebra operations
have been put forward by Rost, Deligne, and Loos. We present a simpler
and more explicit construction that does not break down into cases based
on the parity of $n$. We then prove properties of this construction, and
compute some examples explicitly.

2010 Mathematics Subject Classification: Primary 13B02; Secondary 14B25, 11R11, 13B40, 13C10

Keywords and Phrases: discriminant algebra, discriminant form, algebra of finite rank, étale algebra, polynomial law

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