The Stacks project

Remark 32.6.2. Let $S$ be a scheme. Let us say that a functor $F : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ is limit preserving if for every directed inverse system $\{ T_ i\} _{i \in I}$ of affine schemes with limit $T$ we have $F(T) = \mathop{\mathrm{colim}}\nolimits _ i F(T_ i)$. Let $X$ be a scheme over $S$, and let $h_ X : (\mathit{Sch}/S)^{opp} \to \textit{Sets}$ be its functor of points, see Schemes, Section 26.15. In this terminology Proposition 32.6.1 says that a scheme $X$ is locally of finite presentation over $S$ if and only if $h_ X$ is limit preserving.

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