The Stacks project

Lemma 21.44.5. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $U$ be an object of $\mathcal{C}$. Given a solid diagram of $\mathcal{O}_ U$-modules

\[ \xymatrix{ \mathcal{E} \ar@{..>}[dr] \ar[r] & \mathcal{F} \\ & \mathcal{G} \ar[u]_ p } \]

with $\mathcal{E}$ a direct summand of a finite free $\mathcal{O}_ U$-module and $p$ surjective, then there exists a covering $\{ U_ i \to U\} $ such that a dotted arrow making the diagram commute exists over each $U_ i$.

Proof. We may assume $\mathcal{E} = \mathcal{O}_ U^{\oplus n}$ for some $n$. In this case finding the dotted arrow is equivalent to lifting the images of the basis elements in $\Gamma (U, \mathcal{F})$. This is locally possible by the characterization of surjective maps of sheaves (Sites, Section 7.11). $\square$

Comments (0)

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Your email address will not be published.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 08FN. Beware of the difference between the letter 'O' and the digit '0'.