Lemma 28.23.2 (077M)—The Stacks project

Loading web-font TeX/Main/Regular

Table of contents
Part 2: Schemes
Chapter 28: Properties of Schemes
Section 28.23: Gabber's result
Lemma 28.23.2 (cite)

numberstags

Lemma 28.23.2. 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)

There are also:

4 comment(s) on Section 28.23: Gabber's result

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).

Comments (0)

Post a comment