Proposition 69.3.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

1. The morphism $f$ is a morphism of algebraic spaces which is locally of finite presentation, see Morphisms of Spaces, Definition 66.28.1.

2. The morphism $f : X \to Y$ is limit preserving as a transformation of functors, see Definition 69.3.1.

Proof. Assume (1). Let $T$ be a scheme and let $y \in Y(T)$. We have to show that $T \times _ Y X$ is limit preserving over $T$ in the sense of Definition 69.3.1. Hence we are reduced to proving that if $X$ is an algebraic space which is locally of finite presentation over $S$ as an algebraic space, then it is limit preserving as a functor $X : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$. To see this choose a presentation $X = U/R$, see Spaces, Definition 64.9.3. It follows from Morphisms of Spaces, Definition 66.28.1 that both $U$ and $R$ are schemes which are locally of finite presentation over $S$. Hence by Limits, Proposition 32.6.1 we have

$U(T) = \mathop{\mathrm{colim}}\nolimits U(T_ i), \quad R(T) = \mathop{\mathrm{colim}}\nolimits R(T_ i)$

whenever $T = \mathop{\mathrm{lim}}\nolimits _ i T_ i$ in $(\mathit{Sch}/S)_{fppf}$. It follows that the presheaf

$(\mathit{Sch}/S)_{fppf}^{opp} \longrightarrow \textit{Sets}, \quad W \longmapsto U(W)/R(W)$

is limit preserving. Hence by Lemma 69.3.7 its sheafification $X = U/R$ is limit preserving too.

Assume (2). Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Next, choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. By Lemma 69.3.5 the transformation of functors $V \times _ Y X \to V$ is limit preserving. By Morphisms of Spaces, Lemma 66.39.8 the morphism of algebraic spaces $U \to V \times _ Y X$ is locally of finite presentation, hence limit preserving as a transformation of functors by the first part of the proof. By Lemma 69.3.3 the composition $U \to V \times _ Y X \to V$ is limit preserving as a transformation of functors. Hence the morphism of schemes $U \to V$ is locally of finite presentation by Limits, Proposition 32.6.1 (modulo a set theoretic remark, see last paragraph of the proof). This means, by definition, that (1) holds.

Set theoretic remark. Let $U \to V$ be a morphism of $(\mathit{Sch}/S)_{fppf}$. In the statement of Limits, Proposition 32.6.1 we characterize $U \to V$ as being locally of finite presentation if for all directed inverse systems $(T_ i, f_{ii'})$ of affine schemes over $V$ we have $U(T) = \mathop{\mathrm{colim}}\nolimits V(T_ i)$, but in the current setting we may only consider affine schemes $T_ i$ over $V$ which are (isomorphic to) an object of $(\mathit{Sch}/S)_{fppf}$. So we have to make sure that there are enough affines in $(\mathit{Sch}/S)_{fppf}$ to make the proof work. Inspecting the proof of (2) $\Rightarrow$ (1) of Limits, Proposition 32.6.1 we see that the question reduces to the case that $U$ and $V$ are affine. Say $U = \mathop{\mathrm{Spec}}(A)$ and $V = \mathop{\mathrm{Spec}}(B)$. By construction of $(\mathit{Sch}/S)_{fppf}$ the spectrum of any ring of cardinality $\leq |B|$ is isomorphic to an object of $(\mathit{Sch}/S)_{fppf}$. Hence it suffices to observe that in the "only if" part of the proof of Algebra, Lemma 10.127.3 only $A$-algebras of cardinality $\leq |B|$ are used. $\square$

Comment #2298 by Eric Ahlqvist on

Hi! I think you should switch $X$ and $Y$ in the fiber product on the first line of the proof: $T\times_XY$ should be $T\times_YX$?

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