A new discriminant algebra construction

TL;DR

This paper introduces a simplified, explicit construction of discriminant algebras for commutative rings and their algebras, avoiding case distinctions based on parity, and explores its properties and examples.

Contribution

It provides a new, more straightforward construction of discriminant algebras that is uniform across all ranks, improving upon previous methods.

Findings

The new construction is simpler and more explicit.

It works uniformly for all ranks without parity-based case distinctions.

Several examples are computed explicitly to illustrate the construction.

Abstract

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.