The Stacks project

Lemma 32.8.5. Notation and assumptions as in Situation 32.8.1. If

  1. $f$ is a closed immersion, and

  2. $f_0$ is locally of finite type,

then there exists an $i \geq 0$ such that $f_ i$ is a closed immersion.

Proof. A closed immersion is affine, see Morphisms, Lemma 29.11.9. Hence by Lemma 32.8.2 above after increasing $0$ we may assume that $f_0$ is affine. By writing $Y_0$ as a finite union of affines we reduce to proving the result when $X_0$ and $Y_0$ are affine and map into a common affine $W \subset S_0$. The corresponding algebra statement is a consequence of Algebra, Lemma 10.168.4. $\square$

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