The Stacks project

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 ├ętale 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 ├ętale 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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