Example 59.15.10. Let S be a scheme. Consider the additive group scheme \mathbf{G}_{a, S} = \mathbf{A}^1_ S over S, see Groupoids, Example 39.5.3. The associated representable presheaf is given by
h_{\mathbf{G}_{a, S}}(T) = \mathop{\mathrm{Mor}}\nolimits _ S(T, \mathbf{G}_{a, S}) = \Gamma (T, \mathcal{O}_ T).
By the above we now know that this is a presheaf of sets which satisfies the sheaf condition for the fpqc topology. On the other hand, it is clearly a presheaf of rings as well. Hence we can think of this as a functor
\begin{matrix} \mathcal{O} :
& (\mathit{Sch}/S)^{opp}
& \longrightarrow
& \textit{Rings}
\\ & T/S
& \longmapsto
& \Gamma (T, \mathcal{O}_ T)
\end{matrix}
which satisfies the sheaf condition for the fpqc topology. Correspondingly there is a notion of \mathcal{O}-module, and so on and so forth.
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