The Stacks project

63.7 Fibre products of algebraic spaces

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

Lemma 63.7.1. 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 63.3.4. 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 63.5.7. $\square$

Lemma 63.7.2. 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 63.6.1. 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 63.3.4 and 63.3.3. 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 63.3.2 and 63.3.3. This proves the last condition of Definition 63.6.1 holds and we conclude that $F \times _ H G$ is an algebraic space. $\square$

Lemma 63.7.3. 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 63.7.2 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 $(\mathit{Sch}/S)_{fppf}$. $\square$


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 04T8. Beware of the difference between the letter 'O' and the digit '0'.