Lemma 29.16.6. Let $f : T \to S$ be a morphism of schemes. If $f$ is locally of finite type and surjective, then $f(T_{\text{ft-pts}}) = S_{\text{ft-pts}}$.
Proof. We have $f(T_{\text{ft-pts}}) \subset S_{\text{ft-pts}}$ by Lemma 29.16.5. Let $s \in S$ be a finite type point. As $f$ is surjective the scheme $T_ s = \mathop{\mathrm{Spec}}(\kappa (s)) \times _ S T$ is nonempty, therefore has a finite type point $t \in T_ s$ by Lemma 29.16.4. Now $T_ s \to T$ is a morphism of finite type as a base change of $s \to S$ (Lemma 29.15.4). Hence the image of $t$ in $T$ is a finite type point by Lemma 29.16.5 which maps to $s$ by construction. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)