The Stacks project

Lemma 96.5.5. Let $S$ be a scheme. Let $\kappa = \text{size}(T)$ for some $T \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$. Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ such that

  1. $\mathcal{Y} \to (\mathit{Sch}/S)_{fppf}$ is limit preserving on objects,

  2. for an affine scheme $V$ locally of finite presentation over $S$ and $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ V)$ the fibre product $(\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is representable by an algebraic space of size $\leq \kappa $1,

  3. $\mathcal{X}$ and $\mathcal{Y}$ are stacks for the Zariski topology.

Then $f$ is representable by algebraic spaces.

Proof. Let $V$ be a scheme over $S$ and $y \in \mathcal{Y}_ V$. We have to prove $(\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is representable by an algebraic space.

Case I: $V$ is affine and maps into an affine open $\mathop{\mathrm{Spec}}(\Lambda ) \subset S$. Then we can write $V = \mathop{\mathrm{lim}}\nolimits V_ i$ with each $V_ i$ affine and of finite presentation over $\mathop{\mathrm{Spec}}(\Lambda )$, see Algebra, Lemma 10.127.2. Then $y$ comes from an object $y_ i$ over $V_ i$ for some $i$ by assumption (1). By assumption (3) the fibre product $(\mathit{Sch}/V_ i)_{fppf} \times _{y_ i, \mathcal{Y}} \mathcal{X}$ is representable by an algebraic space $Z_ i$. Then $(\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is representable by $Z \times _{V_ i} V$.

Case II: $V$ is general. Choose an affine open covering $V = \bigcup _{i \in I} V_ i$ such that each $V_ i$ maps into an affine open of $S$. We first claim that $\mathcal{Z} = (\mathit{Sch}/V)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X}$ is a stack in setoids for the Zariski topology. Namely, it is a stack in groupoids for the Zariski topology by Stacks, Lemma 8.5.6. Then suppose that $z$ is an object of $\mathcal{Z}$ over a scheme $T$. Denote $g : T \to V$ the morphism corresponding to the projection of $z$ in $(\mathit{Sch}/V)_{fppf}$. Consider the Zariski sheaf $\mathit{I} = \mathit{Isom}_{\mathcal{Z}}(z, z)$. By Case I we see that $\mathit{I}|_{g^{-1}(V_ i)} = *$ (the singleton sheaf). Hence $\mathcal{I} = *$. Thus $\mathcal{Z}$ is fibred in setoids. To finish the proof we have to show that the Zariski sheaf $Z : T \mapsto \mathop{\mathrm{Ob}}\nolimits (\mathcal{Z}_ T)/\cong $ is an algebraic space, see Algebraic Stacks, Lemma 93.8.2. There is a map $p : Z \to V$ (transformation of functors) and by Case I we know that $Z_ i = p^{-1}(V_ i)$ is an algebraic space. The morphisms $Z_ i \to Z$ are representable by open immersions and $\coprod Z_ i \to Z$ is surjective (in the Zariski topology). Hence $Z$ is a sheaf for the fppf topology by Bootstrap, Lemma 79.3.11. Thus Spaces, Lemma 64.8.5 applies and we conclude that $Z$ is an algebraic space2. $\square$

[1] The condition on size can be dropped by those ignoring set theoretic issues.
[2] To see that the set theoretic condition of that lemma is satisfied we argue as follows: First choose the open covering such that $|I| \leq \text{size}(V)$. Next, choose schemes $U_ i$ of size $\leq \max (\kappa , \text{size}(V))$ and surjective ├ętale morphisms $U_ i \to Z_ i$; we can do this by assumption (2) and Sets, Lemma 3.9.6 (details omitted). Then Sets, Lemma 3.9.9 implies that $\coprod U_ i$ is an object of $(\mathit{Sch}/S)_{fppf}$. Hence $\coprod Z_ i$ is an algebraic space by Spaces, Lemma 64.8.4.

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