A new discriminant algebra construction (Q321892)

Summary: 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 \textit{M. Rost} [``The discriminant algebra of a cubic algebra'', Preprint, \url{https://www.math.uni-bielefeld.de/~rost/data/cub-disc.pdf}; \textit{P. Deligne}, ``Letter to M. Rost and M. Bhargava'' (2005); \textit{O. Loos}, Monatsh. Math. 122, No. 1, 45--98 (1996; Zbl 0872.11022)]. 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.