The Stacks project

Remark 66.3.9. Here is an important special case of Proposition 66.3.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Then $X$ is locally of finite presentation over $S$ if and only if $X$, as a functor $(\mathit{Sch}/S)^{opp} \to \textit{Sets}$, is limit preserving. Compare with Limits, Remark 31.6.2. In fact, we will see in Lemma 66.3.10 below that it suffices if the map

\[ \mathop{\mathrm{colim}}\nolimits X(T_ i) \longrightarrow X(T) \]

is surjective whenever $T = \mathop{\mathrm{lim}}\nolimits T_ i$ is a directed limit of affine schemes over $S$.


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