The Stacks project

Definition 69.3.1. 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 $G$.

[1] The characterization (2) in Lemma 69.3.2 may be easier to parse.

Comments (2)

Comment #3184 by A. on

In (1) of the definition, the formulation says that we need to check "equality" F(lim T_i)=colim F(T_i) for all systems T_i but only such that the limit lim T_i is affine over S. This is somewhat confusing as this last condition seems to be always satisfied.

Comment #3295 by on

Yes, it is true that an inverse system of affine schemes (over a base) always has a limit in the category of schemes (over the base) and that this limit is an affine scheme (over the base). On the other hand, the current formulation in part (1) of the definition, doesn't require the reader to know that this is true. So it would only be confusing to readers who "know too much", such as yourself.

Still, I'll change it if somebody gives me a good alternative formulation.


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