The Stacks project

Lemma 70.18.9. Let $S$ be a scheme. Let $B$ be an algebraic space over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ B$-module. Let $Z_ k \subset S$ be the closed subscheme cut out by $\text{Fit}_ k(\mathcal{F})$, see Section 70.5. Let $B' \to B$ be the blowup of $B$ in $Z_ k$ and let $\mathcal{F}'$ be the strict transform of $\mathcal{F}$. Then $\mathcal{F}'$ can locally be generated by $\leq k$ sections.

Proof. Omitted. Follows from the case of schemes (Divisors, Lemma 31.35.1) by ├ętale localization (Lemma 70.18.2). $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 70.18: Strict transform

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Your email address will not be published.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 0D0P. Beware of the difference between the letter 'O' and the digit '0'.