A new discriminant algebra construction

Abstract

A discriminant algebra operation sends a commutative ring R and an R-algebra A of rank n to an R-algebra ∆ 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.

Discriminants and the monoid of quadratic rings

“…Recently, there has been renewed interest in the construction of discriminant algebras (sending an R-algebra A of rank n to a quadratic R-algebra) by Loos [15], Rost [17], and more recently by Biesel and Gioia [4]. Indeed, Biesel and Gioia [4, Section 8] describe the monoid operation in Theorem A over an affine base in the context of discriminant algebras.…”

“…Recalling the case of quadratic extensions of a field F with char F = 2, for a commutative ring R we define the Artin-Schreier group AS(R) to be the additive quotient [4] where…”

“…Theorem C. The fibers of the map disc : Quad(R) → Disc(R) have a unique action of the group AS(R) compatible with the inclusion of monoids AS(R) ֒→ Quad(R). Moreover, the kernel of this action on the fiber disc −1 (dR ×2 ) contains ann R (d) [4].…”

“…. If 2 is invertible on X, then X [4] is the empty scheme and the square condition (2.11) is vacuously satisfied. Definition 2.12.…”

“…, where [4] denotes working modulo 4, as in the previous section. We adapt the argument in Lemma 3.5 to a global setting.…”