The Stacks project

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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View Example 59.15.10 as pdf