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Working over an arbitrary base scheme $S$, we define the canonical quadraticpair on the Clifford algebra associated to an Azumaya algebra with quadraticpair. Given an Azumaya algebra $\mathcal{A}$ with quadratic pair, i.e., with anorthogonal involution and a semi-trace, its associated Clifford algebra'scanonical 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 quadraticpair on the Clifford algebra is extended from previous work of Dolphin andQu\'eguiner-Mathieu, who worked over fields of characteristic $2$. When$\mathrm{deg}(\mathcal{A})=4$, we show that no canonical quadratic pair exists.

The Canonical Quadratic Pair on Clifford Algebras over Schemes