Lemma 31.9.4. Let $S$ be a scheme. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ S$-module. Let $s \in S$. Then $\mathcal{F}$ can be generated by $r$ elements in a neighbourhood of $s$ if and only if $\text{Fit}_ r(\mathcal{F})_ s = \mathcal{O}_{S, s}$.

**Proof.**
Follows immediately from More on Algebra, Lemma 15.8.7.
$\square$

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