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. $\square$

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