The Stacks project

Lemma 66.3.4. Let $S$ be a scheme. Let $g : Y \to X$ be a representable morphism of algebraic spaces over $S$. Let $f : U \to X$ be a morphism from a scheme towards $X$. Let $f' : U \times _ X Y \to Y$ be the base change of $f$. If

\[ \mathop{\mathrm{Im}}(|f| : |U| \to |X|) \subset \mathop{\mathrm{Im}}(|g| : |Y| \to |X|) \]

and $f'$ has universally bounded fibres, then $f$ has universally bounded fibres.

Proof. Let $n \geq 0$ be an integer bounding the degrees of the fibre products $\mathop{\mathrm{Spec}}(k) \times _ Y (U \times _ X Y)$ as in Definition 66.3.1 for the morphism $f'$. We claim that $n$ works for $f$ also. Namely, suppose that $x : \mathop{\mathrm{Spec}}(k) \to X$ is a morphism from the spectrum of a field. Then either $\mathop{\mathrm{Spec}}(k) \times _ X U$ is empty (and there is nothing to prove), or $x$ is in the image of $|f|$. By Properties of Spaces, Lemma 64.4.3 and the assumption of the lemma we see that this means there exists a field extension $k \subset k'$ and a commutative diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(k') \ar[r] \ar[d] & Y \ar[d] \\ \mathop{\mathrm{Spec}}(k) \ar[r] & X } \]

Hence we see that

\[ \mathop{\mathrm{Spec}}(k') \times _ Y (U \times _ X Y) = \mathop{\mathrm{Spec}}(k') \times _{\mathop{\mathrm{Spec}}(k)} (\mathop{\mathrm{Spec}}(k) \times _ X U) \]

Since the scheme $\mathop{\mathrm{Spec}}(k') \times _ Y (U \times _ X Y)$ is assumed finite of degree $\leq n$ over $k'$ it follows that also $\mathop{\mathrm{Spec}}(k) \times _ X U$ is finite of degree $\leq n$ over $k$ as desired. (Some details omitted.) $\square$


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Your email address will not be published.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 03JO. Beware of the difference between the letter 'O' and the digit '0'.