On the classification of quadratic forms over an integral domain of a global function field

Let $C$ be a smooth projective curve defined over the finite field ${\mathbb{F}}_q$ ($q$ is odd) and let $K={\mathbb{F}}_q(C)$ be its function field. Any finite set $S$ of closed points of $C$ gives rise to an integral domain ${\mathcal{O}}_S:={\mathbb{F}}_q[C-S]$ in $K$. We show that given an ${\mathcal{O}}_S$-regular quadratic space $(V,q)$ of rank $n\ge 3$, the set of genera in the proper classification of quadratic ${\mathcal{O}}_S$-spaces isomorphic to $(V,q)$ in the flat or 'etale topology, is in $1:1$ correspondence with ${}_2\text{Br}({\mathcal{O}}_S)$, thus there are $2^{\mathrm{\mid }S\mathrm{\mid }-1}$ such. If $(V,q)$ is isotropic, then $\text{Pic}({\mathcal{O}}_S)\mathrm{/}2$ classifies the forms in the genus of $(V,q)$. For $n\ge 5$ this is true for all genera, hence the full classification is via the abelian group $H_{\acute{\text{e}}\text{t}}^2({\mathcal{O}}_S,{\underset{‾}{\mu }}_2)$.