Lemma 70.5.4. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type, quasi-coherent $\mathcal{O}_ X$-module. Let $x \in |X|$. Then $\mathcal{F}$ can be generated by $r$ elements in an étale neighbourhood of $x$ if and only if $\text{Fit}_ r(\mathcal{F})_{\overline{x}} = \mathcal{O}_{X, \overline{x}}$.

**Proof.**
Reduces to Divisors, Lemma 31.9.4 by étale localization (as well as the description of the local ring in Properties of Spaces, Section 65.22 and the fact that the strict henselization of a local ring is faithfully flat to see that the equality over the strict henselization is equivalent to the equality over the local ring).
$\square$

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