The Stacks project

Definition 59.17.2. Let $\mathcal{C}$ be a ringed site, i.e., a site endowed with a sheaf of rings $\mathcal{O}$. A sheaf of $\mathcal{O}$-modules $\mathcal{F}$ on $\mathcal{C}$ is called quasi-coherent if for all $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a covering $\{ U_ i \to U\} _{i\in I}$ of $\mathcal{C}$ such that the restriction $\mathcal{F}|_{\mathcal{C}/U_ i}$ is isomorphic to the cokernel of an $\mathcal{O}$-linear map of free $\mathcal{O}$-modules

\[ \bigoplus \nolimits _{k \in K} \mathcal{O}|_{\mathcal{C}/U_ i} \longrightarrow \bigoplus \nolimits _{l \in L} \mathcal{O}|_{\mathcal{C}/U_ i}. \]

The direct sum over $K$ is the sheaf associated to the presheaf $V \mapsto \bigoplus _{k \in K} \mathcal{O}(V)$ and similarly for the other.


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Your email address will not be published.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 03OH. Beware of the difference between the letter 'O' and the digit '0'.