The Stacks project

85.10 Fibre products

Obligatory section about fibre products of formal algebraic spaces.

Lemma 85.10.1. Let $S$ be a scheme. Let $\{ X_ i \to X\} _{i \in I}$ be a family of maps of sheaves on $(\mathit{Sch}/S)_{fppf}$. Assume (a) $X_ i$ is a formal algebraic space over $S$, (b) $X_ i \to X$ is representable by algebraic spaces and ├ętale, and (c) $\coprod X_ i \to X$ is a surjection of sheaves. Then $X$ is a formal algebraic space over $S$.

Proof. For each $i$ pick $\{ X_{ij} \to X_ i\} _{j \in J_ i}$ as in Definition 85.7.1. Then $\{ X_{ij} \to X\} _{i \in I, j \in J_ i}$ is a family as in Definition 85.7.1 for $X$. $\square$

Lemma 85.10.2. Let $S$ be a scheme. Let $X, Y$ be formal algebraic spaces over $S$ and let $Z$ be a sheaf whose diagonal is representable by algebraic spaces. Let $X \to Z$ and $Y \to Z$ be maps of sheaves. Then $X \times _ Z Y$ is a formal algebraic space.

Proof. Choose $\{ X_ i \to X\} $ and $\{ Y_ j \to Y\} $ as in Definition 85.7.1. Then $\{ X_ i \times _ Z Y_ j \to X \times _ Z Y\} $ is a family of maps which are representable by algebraic spaces and ├ętale. Thus Lemma 85.10.1 tells us it suffices to show that $X \times _ Z Y$ is a formal algebraic space when $X$ and $Y$ are affine formal algebraic spaces.

Assume $X$ and $Y$ are affine formal algebraic spaces. Write $X = \mathop{\mathrm{colim}}\nolimits X_\lambda $ and $Y = \mathop{\mathrm{colim}}\nolimits Y_\mu $ as in Definition 85.5.1. Then $X \times _ Z Y = \mathop{\mathrm{colim}}\nolimits X_\lambda \times _ Z Y_\mu $. Each $X_\lambda \times _ Z Y_\mu $ is an algebraic space. For $\lambda \leq \lambda '$ and $\mu \leq \mu '$ the morphism

\[ X_\lambda \times _ Z Y_\mu \to X_\lambda \times _ Z Y_{\mu '} \to X_{\lambda '} \times _ Z Y_{\mu '} \]

is a thickening as a composition of base changes of thickenings. Thus we conclude by applying Lemma 85.8.1. $\square$

Lemma 85.10.3. Let $S$ be a scheme. The category of formal algebraic spaces over $S$ has fibre products.

Proof. Special case of Lemma 85.10.2 because formal algebraic spaces have representable diagonals, see Lemma 85.7.3. $\square$

Lemma 85.10.4. Let $S$ be a scheme. Let $X \to Z$ and $Y \to Z$ be morphisms of formal algebraic spaces over $S$. Then $(X \times _ Z Y)_{red} = (X_{red} \times _{Z_{red}} Y_{red})_{red}$.

Proof. This follows from the universal property of the reduction in Lemma 85.7.2. $\square$

We have already proved the following lemma (without knowing that fibre products exist).

Lemma 85.10.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The diagonal morphism $\Delta : X \to X \times _ Y X$ is representable (by schemes), a monomorphism, locally quasi-finite, locally of finite type, and separated.

Proof. Let $T$ be a scheme and let $T \to X \times _ Y X$ be a morphism. Then

\[ T \times _{(X \times _ Y X)} X = T \times _{(X \times _ S X)} X \]

Hence the result follows immediately from Lemma 85.7.3. $\square$


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