Lemma 31.35.1 (0CZP)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.35: Blowing up and flatness
Lemma 31.35.1 (cite)

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Lemma 31.35.1. 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. Let $S' \to S$ be the blowup of $S$ in $Z_ k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$ . Then $\mathcal{F}'$ can locally be generated by $\leq k$ sections.

Proof. Recall that $\mathcal{F}'$ can locally be generated by $\leq k$ sections if and only if $\text{Fit}_ k(\mathcal{F}') = \mathcal{O}_{S'}$ , see Lemma 31.9.4. Hence this lemma is a translation of More on Algebra, Lemma 15.26.3. $\square$

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