Lemma 38.37.2 (0EV8)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.37: Almost blow up squares and the h topology
Lemma 38.37.2 (cite)

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Lemma 38.37.2. Consider an almost blow up square (38.37.0.1). Let $W \to X'$ be a closed immersion of finite presentation. The following are equivalent

$X' \setminus E$ is scheme theoretically contained in $W$,
the blowup $X''$ of $X$ in $Z$ is scheme theoretically contained in $W$,
the diagram

\[ \xymatrix{ E \cap W \ar[d] \ar[r] & W \ar[d] \\ Z \ar[r] & X } \]

is an almost blow up square. Here $E \cap W$ is the scheme theoretic intersection.

Proof. Assume (1). Then the surjection $\mathcal{O}_{X'} \to \mathcal{O}_ W$ is an isomorphism over the open $X' \subset E$. Since the ideal sheaf of $X'' \subset X'$ is the sections of $\mathcal{O}_{X'}$ supported on $E$ (by our definition of almost blow up squares) we conclude (2) is true. If (2) is true, then (3) holds. If (3) holds, then (1) holds because $X'' \cap (X' \setminus E)$ is isomorphic to $X \setminus Z$ which in turn is isomorphic to $X' \setminus E$. $\square$

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