The Stacks project

Definition 70.3.1. Let $S$ be a scheme.

1. A functor $F : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ is said to be limit preserving or locally of finite presentation if for every affine scheme $T$ over $S$ which is a limit $T = \mathop{\mathrm{lim}}\nolimits T_ i$ of a directed inverse system of affine schemes $T_ i$ over $S$, we have

   $$F(T) = \mathop{\mathrm{colim}}\nolimits F(T_ i).$$

   We sometimes say that $F$ is locally of finite presentation over $S$.

2. Let $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. A transformation of functors $a : F \to G$ is limit preserving or locally of finite presentation if for every scheme $T$ over $S$ and every $y \in G(T)$ the functor

   $$F_ y : (\mathit{Sch}/T)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad T'/T \longmapsto \{ x \in F(T') \mid a(x) = y|_{T'}\}$$

   is locally of finite presentation over $T$¹. We sometimes say that $F$ is relatively limit preserving over $G$.