Lemma 31.9.5 (0C3G)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.9: Fitting ideals
Lemma 31.9.5 (cite)

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Lemma 31.9.5. Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ S$ -module. Let $r \geq 0$ . The following are equivalent

$\mathcal{F}$ is finite locally free of rank $r$
$\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and $\text{Fit}_ r(\mathcal{F}) = \mathcal{O}_ S$ , and
$\text{Fit}_ k(\mathcal{F}) = 0$ for $k < r$ and $\text{Fit}_ k(\mathcal{F}) = \mathcal{O}_ S$ for $k \geq r$ .

Proof. Follows immediately from More on Algebra, Lemma 15.8.8. $\square$

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