The Canonical Quadratic Pair on Clifford Algebras over Schemes

TL;DR

This paper extends the definition of the canonical quadratic pair on Clifford algebras associated with Azumaya algebras with quadratic pairs over arbitrary schemes, clarifying when the involution is orthogonal and establishing non-existence in degree 4.

Contribution

It generalizes previous work by defining the canonical quadratic pair over schemes and characterizes its properties depending on the algebra's degree and base scheme characteristics.

Findings

Canonical quadratic pair defined for degrees ≥ 8

Involution orthogonal only under specific conditions

No canonical quadratic pair exists for degree 4

Abstract

Working over an arbitrary base scheme $S$, we define the canonical quadratic pair on the Clifford algebra associated to an Azumaya algebra with quadratic pair. Given an Azumaya algebra $\mathcal{A}$ with quadratic pair, i.e., with an orthogonal involution and a semi-trace, its associated Clifford algebra's canonical involution is only orthogonal in certain cases, namely when $\mathrm{deg}(\mathcal{A})$ is divisible by $8$ or when both $2=0$ over $S$ and $\mathrm{deg}(\mathcal{A})$ is divisible by $4$. When $\mathrm{deg}(\mathcal{A}) \geq 8$, our definition of the canonical quadratic pair on the Clifford algebra is extended from previous work of Dolphin and Qu\'eguiner-Mathieu, who worked over fields of characteristic $2$. When $\mathrm{deg}(\mathcal{A})=4$, we show that no canonical quadratic pair exists.