The Stacks project

Lemma 75.49.5. Consider a commutative diagram

\[ \xymatrix{ X \ar[rr]_ f \ar[rd]_ p & & Y \ar[ld]^ q \\ & Z } \]

of algebraic spaces. Assume that

  1. $p$ is locally of finite presentation,

  2. $p$ is flat,

  3. $p$ is closed,

  4. $q$ is locally of finite type, and

  5. $q$ is closed.

Then there exists an open subspace $W \subset Z$ such that a morphism $Z' \to Z$ factors through $W$ if and only if the base change $f_{Z'} : X_{Z'} \to Y_{Z'}$ is surjective and flat.

Proof. By Lemma 75.49.4 we may assume that $f$ is flat. Note that $f$ is locally of finite presentation by Morphisms of Spaces, Lemma 66.28.9. Hence $f$ is open, see Morphisms of Spaces, Lemma 66.30.6. Let $W \subset Z$ be the open subspace (see Properties of Spaces, Lemma 65.4.8) with underlying set of points

\[ |W| = |Z| \setminus |q|\left(|Y| \setminus |f|(|X|)\right). \]

in other words for $z \in |Z|$ we have $z \in |W|$ if and only if the whole fibre of $|Y| \to |Z|$ over $z$ is in the image of $|X| \to |Y|$. Since $q$ is closed this set is open in $|Z|$. The morphism $X_ W \to Y_ W$ is surjective by construction. Finally, suppose that $X_{Z'} \to Y_{Z'}$ is surjective. In order to show that $Z' \to Z$ factors through $W$ it suffices to show that $|Z'| \to |Z|$ has image contained in $|W|$, see Properties of Spaces, Lemma 65.4.9. Hence it suffices to prove the result when $Z'$ is the spectrum of a field. Denote $z \in |Z|$ the image of $|Z'| \to |Z|$. By Properties of Spaces, Lemma 65.4.3 in the commutative diagram

\[ \xymatrix{ |X_{Z'}| \ar[d] \ar[r] & |Y_{Z'}| \ar[d] \\ |p|^{-1}(\{ z\} ) \ar[r] & |q|^{-1}(\{ z\} ) } \]

the vertical arrows are surjective. It follows that $z \in |W|$ as desired. $\square$


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Your email address will not be published.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 05XC. Beware of the difference between the letter 'O' and the digit '0'.