The Stacks project

Lemma 31.35.2. Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ S$-module. Let $Z_ k \subset S$ be the closed subscheme cut out by $\text{Fit}_ k(\mathcal{F})$, see Section 31.9. Assume that $\mathcal{F}$ is locally free of rank $k$ on $S \setminus Z_ k$. Let $S' \to S$ be the blowup of $S$ in $Z_ k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ is locally free of rank $k$.

Proof. Translation of More on Algebra, Lemma 15.26.4. $\square$

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