The Stacks project

64.7 Fibre products of algebraic spaces

The category of algebraic spaces over $S$ has both products and fibre products.

Lemma 64.7.1. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F, G$ be algebraic spaces over $S$. Then $F \times G$ is an algebraic space, and is a product in the category of algebraic spaces over $S$.

Proof. It is clear that $H = F \times G$ is a sheaf. The diagonal of $H$ is simply the product of the diagonals of $F$ and $G$. Hence it is representable by Lemma 64.3.4. Finally, if $U \to F$ and $V \to G$ are surjective étale morphisms, with $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$, then $U \times V \to F \times G$ is surjective étale by Lemma 64.5.7. $\square$

Lemma 64.7.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $H$ be a sheaf on $(\mathit{Sch}/S)_{fppf}$ whose diagonal is representable. Let $F, G$ be algebraic spaces over $S$. Let $F \to H$, $G \to H$ be maps of sheaves. Then $F \times _ H G$ is an algebraic space.

Proof. We check the 3 conditions of Definition 64.6.1. A fibre product of sheaves is a sheaf, hence $F \times _ H G$ is a sheaf. The diagonal of $F \times _ H G$ is the left vertical arrow in

\[ \xymatrix{ F \times _ H G \ar[r] \ar[d]_\Delta & F \times G \ar[d]^{\Delta _ F \times \Delta _ G} \\ (F \times F) \times _{(H \times H)} (G \times G) \ar[r] & (F \times F) \times (G \times G) } \]

which is cartesian. Hence $\Delta $ is representable as the base change of the morphism on the right which is representable, see Lemmas 64.3.4 and 64.3.3. Finally, let $U, V \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and $a : U \to F$, $b : V \to G$ be surjective and étale. As $\Delta _ H$ is representable, we see that $U \times _ H V$ is a scheme. The morphism

\[ U \times _ H V \longrightarrow F \times _ H G \]

is surjective and étale as a composition of the base changes $U \times _ H V \to U \times _ H G$ and $U \times _ H G \to F \times _ H G$ of the étale surjective morphisms $U \to F$ and $V \to G$, see Lemmas 64.3.2 and 64.3.3. This proves the last condition of Definition 64.6.1 holds and we conclude that $F \times _ H G$ is an algebraic space. $\square$

Lemma 64.7.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $F \to H$, $G \to H$ be morphisms of algebraic spaces over $S$. Then $F \times _ H G$ is an algebraic space, and is a fibre product in the category of algebraic spaces over $S$.

Proof. It follows from the stronger Lemma 64.7.2 that $F \times _ H G$ is an algebraic space. It is clear that $F \times _ H G$ is a fibre product in the category of algebraic spaces over $S$ since that is a full subcategory of the category of (pre)sheaves of sets on $(\mathit{Sch}/S)_{fppf}$. $\square$


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