Lemma 38.23.4 (07AI)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.23: Flattening a map
Lemma 38.23.4 (cite)

previous theorem

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Lemma 38.23.4. Let $f:X\to S$ be a morphism of schemes which is of finite presentation, flat, and pure. Let $Y$ be a closed subscheme of $X$ . Let $F=f_*Y$ be the Weil restriction functor of $Y$ along $f$ , defined by

$F : (\mathit{Sch}/S)^{opp} \to \textit{Sets}, \quad T \mapsto \left\{ \begin{matrix} \{ *\} & \text{if} & Y_ T\to X_ T \text{ is an isomorphism, } \\ \emptyset & \text{else.} & \end{matrix} \right.$

Then $F$ is representable by a closed immersion $Z\to S$ . Moreover $Z\to S$ is of finite presentation if $Y\to S$ is.

Proof. Let $\mathcal{I}$ be the ideal sheaf defining $Y$ in $X$ and let $u:\mathcal{O}_ X\to \mathcal{O}_ X/\mathcal{I}$ be the surjection. Then for an $S$ -scheme $T$ , the closed immersion $Y_ T\to X_ T$ is an isomorphism if and only if $u_ T$ is an isomorphism. Hence the result follows from Theorem 38.23.3. $\square$

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