Definition 69.3.1. Let $S$ be a scheme.

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 $S$*.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 $G$.

## Comments (2)

Comment #3184 by A. on

Comment #3295 by Johan on