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The Stacks project

Lemma 71.18.10. Let S be a scheme. Let B be an algebraic space over S. Let \mathcal{F} be a finite type quasi-coherent \mathcal{O}_ B-module. Let Z_ k \subset S be the closed subscheme cut out by \text{Fit}_ k(\mathcal{F}), see Section 71.5. Assume that \mathcal{F} is locally free of rank k on B \setminus Z_ k. Let B' \to B be the blowup of B in Z_ k and let \mathcal{F}' be the strict transform of \mathcal{F}. Then \mathcal{F}' is locally free of rank k.

Proof. Omitted. Follows from the case of schemes (Divisors, Lemma 31.35.2) by étale localization (Lemma 71.18.2). \square


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