The Stacks project

Lemma 70.3.12. 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 70.3.10 part (2) it suffices to check surjectivity in the criterion of Lemma 70.3.2.

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 32.6.3 (modulo a set theoretic remark exactly as in the proof of Proposition 70.3.10). 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 70.3.10. 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 $b_ i = (U \to V) \circ a_ i$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CM6. Beware of the difference between the letter 'O' and the digit '0'.