Lemma 31.34.2 (080L)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.34: Admissible blowups
Lemma 31.34.2 (cite)

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Admissible blowups are stable under composition.

Lemma 31.34.2. Let $X$ be a quasi-compact and quasi-separated scheme. Let $U \subset X$ be a quasi-compact open subscheme. Let $b : X' \to X$ be a $U$ -admissible blowup. Let $X'' \to X'$ be a $U$ -admissible blowup. Then the composition $X'' \to X$ is a $U$ -admissible blowup.

Proof. Immediate from the more precise Lemma 31.32.14. $\square$

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Comment #833 by Johan Commelin on July 24, 2014 at 18:09

Suggested slogan: Admissible blowups of quasi-compact quasi-separated schemes are stable under composition.

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