The Stacks project

Lemma 70.6.7. Notation and assumptions as in Situation 70.6.1. If

  1. $f$ is finite,

  2. $f_0$ is locally of finite type,

then $f_ i$ is finite for some $i \geq 0$.

Proof. Choose an affine scheme $V_0$ and a surjective ├ętale morphism $V_0 \to Y_0$. Set $V_ i = V_0 \times _{Y_0} Y_ i$ and $V = V_0 \times _{Y_0} Y$. Since $f$ is finite we see that $V \times _ Y X = \mathop{\mathrm{lim}}\nolimits V_ i \times _{Y_ i} X_ i$ is a scheme finite over $V$. By Lemma 70.5.10 we see that $V_ i \times _{Y_ i} X_ i$ is affine for some $i \geq 0$. Increasing $i$ if necessary we find that $V_ i \times _{Y_ i} X_ i \to V_ i$ is finite by Limits, Lemma 32.8.3. For this $i$ the morphism $f_ i$ is finite (Morphisms of Spaces, Lemma 67.45.3). $\square$

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