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The Stacks project

Lemma 70.6.8. Notation and assumptions as in Situation 70.6.1. If

  1. f is a closed immersion,

  2. f_0 is locally of finite type,

then f_ i is a closed immersion 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 a closed immersion we see that V \times _ Y X = \mathop{\mathrm{lim}}\nolimits V_ i \times _{Y_ i} X_ i is a closed subscheme of the affine scheme 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 a closed immersion by Limits, Lemma 32.8.5. For this i the morphism f_ i is a closed immersion (Morphisms of Spaces, Lemma 67.45.3). \square


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