## Tag `0CM6`

Chapter 58: Limits of Algebraic Spaces > Section 58.3: Morphisms of finite presentation

Lemma 58.3.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. If for every directed limit $T = \mathop{\rm lim}\nolimits_{i \in I} T_i$ of affine schemes over $S$ the map $$ \mathop{\rm colim}\nolimits X(T_i) \longrightarrow X(T) \times_{Y(T)} \mathop{\rm colim}\nolimits Y(T_i) $$ is surjective, then $f$ is locally of finite presentation. In other words, in Proposition 58.3.9 part (2) it suffices to check surjectivity in the criterion of Lemma 58.3.2.

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 $T = \mathop{\rm lim}\nolimits T_i$ as in the lemma that the map $$ \mathop{\rm colim}\nolimits U(T_i) \longrightarrow U(T) \times_{V(T)} \mathop{\rm colim}\nolimits V(T_i) $$ 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 58.3.9). 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 $$ \mathop{\rm colim}\nolimits U(T_i) \longrightarrow U(T) \times_{(X \times_Y V)(T)} \mathop{\rm colim}\nolimits (X \times_Y V)(T_i) $$ by Proposition 58.3.9. 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 $b_i = (U \to V) \circ a_i$. $\square$

The code snippet corresponding to this tag is a part of the file `spaces-limits.tex` and is located in lines 554–568 (see updates for more information).

```
\begin{lemma}
\label{lemma-surjection-is-enough}
Let $S$ be a scheme.
Let $f : X \to Y$ be a morphism of algebraic spaces over $S$.
If for every directed limit $T = \lim_{i \in I} T_i$
of affine schemes over $S$ the map
$$
\colim X(T_i) \longrightarrow X(T) \times_{Y(T)} \colim Y(T_i)
$$
is surjective, then $f$ is locally of finite presentation.
In other words, in
Proposition \ref{proposition-characterize-locally-finite-presentation}
part (2) it suffices to check surjectivity in the criterion of
Lemma \ref{lemma-characterize-relative-limit-preserving}.
\end{lemma}
\begin{proof}
Choose a scheme $V$ and a surjective \'etale morphism $g : V \to Y$.
Next, choose a scheme $U$ and a surjective \'etale morphism
$h : U \to V \times_Y X$. It suffices to show for $T = \lim T_i$
as in the lemma that the map
$$
\colim U(T_i) \longrightarrow U(T) \times_{V(T)} \colim V(T_i)
$$
is surjective, because then $U \to V$ will be locally of finite
presentation by Limits, Lemma \ref{limits-lemma-surjection-is-enough}
(modulo a set theoretic remark exactly as in the proof of
Proposition \ref{proposition-characterize-locally-finite-presentation}).
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 \'etale morphism of algebraic spaces
(and hence locally of finite presentation), we have the surjectivity of
$$
\colim U(T_i) \longrightarrow U(T) \times_{(X \times_Y V)(T)}
\colim (X \times_Y V)(T_i)
$$
by Proposition \ref{proposition-characterize-locally-finite-presentation}.
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
$b_i = (U \to V) \circ a_i$.
\end{proof}
```

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