The Stacks project

Lemma 64.3.10. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If for every directed limit $T = \mathop{\mathrm{lim}}\nolimits _{i \in I} T_ i$ of affine schemes over $S$ the map

\[ \mathop{\mathrm{colim}}\nolimits X(T_ i) \longrightarrow X(T) \times _{Y(T)} \mathop{\mathrm{colim}}\nolimits Y(T_ i) \]

is surjective, then $f$ is locally of finite presentation. In other words, in Proposition 64.3.8 part (2) it suffices to check surjectivity in the criterion of Lemma 64.3.2.

Proof. Choose a scheme $V$ and a surjective étale morphism $g : V \to Y$. Next, choose a scheme $U$ and a surjective étale morphism $h : U \to V \times _ Y X$. It suffices to show for $T = \mathop{\mathrm{lim}}\nolimits T_ i$ as in the lemma that the map

\[ \mathop{\mathrm{colim}}\nolimits U(T_ i) \longrightarrow U(T) \times _{V(T)} \mathop{\mathrm{colim}}\nolimits V(T_ i) \]

is surjective, because then $U \to V$ will be locally of finite presentation by Limits, Lemma 31.6.3 (modulo a set theoretic remark exactly as in the proof of Proposition 64.3.8). Thus we take $a : T \to U$ and $b_ i : T_ i \to V$ which determine the same morphism $T \to V$. Picture

\[ \xymatrix{ T \ar[d]_ a \ar[rr]_{p_ i} & & T_ i \ar[d]^{b_ i} \ar@{..>}[ld] \\ U \ar[r]^-h & X \times _ Y V \ar[d] \ar[r] & V \ar[d]^ g \\ & X \ar[r]^ f & Y } \]

By the assumption of the lemma after increasing $i$ we can find a morphism $c_ i : T_ i \to X$ such that $h \circ a = (b_ i, c_ i) \circ p_ i : T_ i \to V \times _ Y X$ and such that $f \circ c_ i = g \circ b_ i$. Since $h$ is an étale morphism of algebraic spaces (and hence locally of finite presentation), we have the surjectivity of

\[ \mathop{\mathrm{colim}}\nolimits U(T_ i) \longrightarrow U(T) \times _{(X \times _ Y V)(T)} \mathop{\mathrm{colim}}\nolimits (X \times _ Y V)(T_ i) \]

by Proposition 64.3.8. Hence after increasing $i$ again we can find the desired morphism $a_ i : T_ i \to U$ with $a = a_ i \circ p_ i$ and $b_ i = (U \to V) \circ a_ i$. $\square$


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