The Stacks project

Lemma 80.11.5. Assume $B \to S$ and $(U, R, s, t, c)$ are as in Groupoids in Spaces, Definition 78.20.1 (1). For any scheme $T$ over $S$ and objects $x, y$ of $[U/R]$ over $T$ the sheaf $\mathit{Isom}(x, y)$ on $(\mathit{Sch}/T)_{fppf}$ is an algebraic space.

Proof. By Groupoids in Spaces, Lemma 78.22.3 there exists an fppf covering $\{ T_ i \to T\} _{i \in I}$ such that $\mathit{Isom}(x, y)|_{(\mathit{Sch}/T_ i)_{fppf}}$ is an algebraic space for each $i$. By Spaces, Lemma 65.16.4 this means that each $F_ i = h_{S_ i} \times \mathit{Isom}(x, y)$ is an algebraic space. Thus to prove the lemma we only have to verify the set theoretic condition that $\coprod F_ i$ is an algebraic space of Lemma 80.11.1 above to conclude. To do this we use Spaces, Lemma 65.8.4 which requires showing that $I$ and the $F_ i$ are not “too large”. We suggest the reader skip the rest of the proof.

Choose $U' \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}/S)_{fppf}$ and a surjective étale morphism $U' \to U$. Let $R'$ be the restriction of $R$ to $U'$. Since $[U/R] = [U'/R']$ we may, after replacing $U$ by $U'$, assume that $U$ is a scheme. (This step is here so that the fibre products below are over a scheme.)

Note that if we refine the covering $\{ T_ i \to T\} $ then it remains true that each $F_ i$ is an algebraic space. Hence we may assume that each $T_ i$ is affine. Since $T_ i \to T$ is locally of finite presentation, this then implies that $\text{size}(T_ i) \leq \text{size}(T)$, see Sets, Lemma 3.9.7. We may also assume that the cardinality of the index set $I$ is at most the cardinality of the set of points of $T$ since to get a covering it suffices to check that each point of $T$ is in the image. Hence $|I| \leq \text{size}(T)$. Choose $W \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and a surjective étale morphism $W \to R$. Note that in the proof of Groupoids in Spaces, Lemma 78.22.3 we showed that $F_ i$ is representable by $T_ i \times _{(y_ i, x_ i), U \times _ B U} R$ for some $x_ i, y_ i : T_ i \to U$. Hence now we see that $V_ i = T_ i \times _{(y_ i, x_ i), U \times _ B U} W$ is a scheme which comes with an étale surjection $V_ i \to F_ i$. By Sets, Lemma 3.9.6 we see that

\[ \text{size}(V_ i) \leq \max \{ \text{size}(T_ i), \text{size}(W)\} \leq \max \{ \text{size}(T), \text{size}(W)\} \]

Hence, by Sets, Lemma 3.9.5 we conclude that

\[ \text{size}(\coprod \nolimits _{i \in I} V_ i) \leq \max \{ |I|, \text{size}(T), \text{size}(W)\} . \]

Hence we conclude by our construction of $\mathit{Sch}$ that $\coprod _{i \in I} V_ i$ is isomorphic to an object $V$ of $(\mathit{Sch}/S)_{fppf}$. This verifies the hypothesis of Spaces, Lemma 65.8.4 and we win. $\square$

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