The Stacks project

Lemma 70.18.8. In the situation of Definition 70.18.1. 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 ├ętale localization (Lemma 70.18.2). $\square$


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