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). $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CZ7. Beware of the difference between the letter 'O' and the digit '0'.