The Stacks project

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. $\square$

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