The Stacks project

Lemma 27.23.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\kappa $ be a cardinal. There exists a set $T$ and a family $(\mathcal{F}_ t)_{t \in T}$ of $\kappa $-generated $\mathcal{O}_ X$-modules such that every $\kappa $-generated $\mathcal{O}_ X$-module is isomorphic to one of the $\mathcal{F}_ t$.

Proof. There is a set of coverings of $X$ (provided we disallow repeats). Suppose $X = \bigcup U_ i$ is a covering and suppose $\mathcal{F}_ i$ is an $\mathcal{O}_{U_ i}$-module. Then there is a set of isomorphism classes of $\mathcal{O}_ X$-modules $\mathcal{F}$ with the property that $\mathcal{F}|_{U_ i} \cong \mathcal{F}_ i$ since there is a set of glueing maps. This reduces us to proving there is a set of (isomorphism classes of) quotients $\oplus _{k \in \kappa } \mathcal{O}_ X \to \mathcal{F}$ for any ringed space $X$. This is clear. $\square$


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Your email address will not be published.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 077M. Beware of the difference between the letter 'O' and the digit '0'.