The Stacks project

Lemma 32.8.13. Notation and assumptions as in Situation 32.8.1. If

  1. $f$ is an immersion, and

  2. $f_0$ is locally of finite type,

then $f_ i$ is an immersion for some $i \geq 0$.

Proof. There exists an open $V \subset Y$ such that the morphism $f$ factors as $X \to V \to Y$ and such that $X \to V$ is a closed immersion, see discussion in Schemes, Section 26.10. Since $X$ is quasi-compact, we may and do assume $V$ is a quasi-compact open of $Y$. By Lemma 32.4.11 after increasing $0$ we can find a quasi-compact open $V_0 \subset Y_0$ such that $V$ is the inverse image of $V_0$. Then the inverse image of $V_0$ in $X_0$ is a quasi-compact open whose inverse image in $X$ is $X$. Hence by the same lemma applied to $X = \mathop{\mathrm{lim}}\nolimits X_ i$ we may assume after increasing $0$ that we have the factorization $X_0 \to V_0 \to Y_0$. Then for large enough $i \geq 0$ the morphism $X_ i \to V_ i$ where $V_ i = Y_ i \times _{Y_0} V_0$ is a closed immersion by Lemma 32.8.5 and the proof is complete. $\square$

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