The Stacks project

Lemma 17.9.8. Let $X$ be a ringed space. There exists a set of $\mathcal{O}_ X$-modules $\{ \mathcal{F}_ i\} _{i \in I}$ of finite type such that each finite type $\mathcal{O}_ X$-module on $X$ is isomorphic to exactly one of the $\mathcal{F}_ i$.

Proof. For each open covering $\mathcal{U} : X = \bigcup U_ j$ consider the sheaves of $\mathcal{O}_ X$-modules $\mathcal{F}$ such that each restriction $\mathcal{F}|_{U_ j}$ is a quotient of $\mathcal{O}_{U_ j}^{\oplus r}$ for some $r_ j \geq 0$. These are parametrized by subsheaves $\mathcal{K}_ i \subset \mathcal{O}_{U_ j}^{\oplus r_ j}$ and glueing data

\[ \varphi _{jj'} : \mathcal{O}_{U_ j \cap U_{j'}}^{\oplus r_ j}/ (\mathcal{K}_ j|_{U_ j \cap U_{j'}}) \longrightarrow \mathcal{O}_{U_ j \cap U_{j'}}^{\oplus r_{j'}}/ (\mathcal{K}_{j'}|_{U_ j \cap U_{j'}}) \]

see Sheaves, Section 6.33. Note that the collection of all glueing data forms a set. The collection of all coverings $\mathcal{U} : X = \bigcup _{j \in J} U_ i$ where $J \to \mathcal{P}(X)$, $j \mapsto U_ j$ is injective forms a set as well. Hence the collection of all sheaves of $\mathcal{O}_ X$-modules gotten from glueing quotients as above forms a set $\mathcal{I}$. By definition every finite type $\mathcal{O}_ X$-module is isomorphic to an element of $\mathcal{I}$. Choosing an element out of each isomorphism class inside $\mathcal{I}$ gives the desired set of sheaves (uses axiom of choice). $\square$


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Your email address will not be published.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 01BC. Beware of the difference between the letter 'O' and the digit '0'.