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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