Lemma 69.6.6. Notation and assumptions as in Situation 69.6.1. If $f$ is affine, then $f_ i$ is affine for some $i \geq 0$.

Proof. Choose an affine scheme $V_0$ and a surjective étale 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 affine we see that $V \times _ Y X = \mathop{\mathrm{lim}}\nolimits V_ i \times _{Y_ i} X_ i$ is affine. By Lemma 69.5.10 we see that $V_ i \times _{Y_ i} X_ i$ is affine for some $i \geq 0$. For this $i$ the morphism $f_ i$ is affine (Morphisms of Spaces, Lemma 66.20.3). $\square$

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