Lemma 32.22.2 (0CNN)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.22: Descending finite type schemes
Lemma 32.22.2 (cite)

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Lemma 32.22.2. In Situation 32.22.1. Let $X \to S$ be quasi-separated and of finite type. Then there exists an $i \in I$ and a diagram

32.22.2.1

$\begin{equation} \label{limits-equation-good-diagram} \vcenter { \xymatrix{ X \ar[r] \ar[d] & W \ar[d] \\ S \ar[r] & S_ i } } \end{equation}$

such that $W \to S_ i$ is of finite type and such that the induced morphism $X \to S \times _{S_ i} W$ is a closed immersion.

Proof. By Lemma 32.9.3 we can find a closed immersion $X \to X'$ over $S$ where $X'$ is a scheme of finite presentation over $S$ . By Lemma 32.10.1 we can find an $i$ and a morphism of finite presentation $X'_ i \to S_ i$ whose pull back is $X'$ . Set $W = X'_ i$ . $\square$

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