The Stacks project

Lemma 95.8.2. There exists a subcategory $\mathcal{S}\! \mathit{paces}_{ft, small} \subset \mathcal{S}\! \mathit{paces}_{ft}$ with the following properties:

  1. the inclusion functor $\mathcal{S}\! \mathit{paces}_{ft, small} \to \mathcal{S}\! \mathit{paces}_{ft}$ is fully faithful and essentially surjective, and

  2. the functor $p_{ft, small} : \mathcal{S}\! \mathit{paces}_{ft, small} \to (\mathit{Sch}/S)_{fppf}$ turns $\mathcal{S}\! \mathit{paces}_{ft, small}$ into a stack over $(\mathit{Sch}/S)_{fppf}$.

Proof. We have seen in Lemmas 95.8.1 that $p_{ft} : \mathcal{S}\! \mathit{paces}_{ft} \to (\mathit{Sch}/S)_{fppf}$ satisfies (1), (2) and (3) of Stacks, Definition 8.4.1. The additional condition (4) of Stacks, Remark 8.4.9 holds because every algebraic space $X$ over $S$ is of the form $U/R$ for $U, R \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, see Spaces, Lemma 65.9.1. Thus there is only a set worth of isomorphism classes of objects. Hence we obtain $\mathcal{S}\! \mathit{paces}_{ft, small}$ from the discussion in that remark. $\square$


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Your email address will not be published.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 04UE. Beware of the difference between the letter 'O' and the digit '0'.