Lemma 32.8.5 (01ZP)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.8: Descending properties of morphisms
Lemma 32.8.5 (cite)

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

$f$ is a closed immersion, and
$f_0$ 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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