The Stacks project

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

  1. $\mathcal{F}$ is finite locally free of rank $r$

  2. $\text{Fit}_{r - 1}(\mathcal{F}) = 0$ and $\text{Fit}_ r(\mathcal{F}) = \mathcal{O}_ S$, and

  3. $\text{Fit}_ k(\mathcal{F}) = 0$ 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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