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 $Δ_{A/R}$ 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.

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