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The Stacks project

Lemma 71.18.8. In the situation of Definition 71.18.1. Suppose that

0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0

is an exact sequence of quasi-coherent sheaves on X which remains exact after any base change T \to B. Then the strict transforms of \mathcal{F}_ i' relative to any blowup B' \to B form a short exact sequence 0 \to \mathcal{F}'_1 \to \mathcal{F}'_2 \to \mathcal{F}'_3 \to 0 too.

Proof. Omitted. Hint: Follows from the case of schemes (Divisors, Lemma 31.33.7) by étale localization (Lemma 71.18.2). \square


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