Lemma 87.15.1. 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 étale, and (c) $\coprod X_ i \to X$ is a surjection of sheaves. Then $X$ is a formal algebraic space over $S$.
87.15 Fibre products
Obligatory section about fibre products of formal algebraic spaces.
Proof. For each $i$ pick $\{ X_{ij} \to X_ i\} _{j \in J_ i}$ as in Definition 87.11.1. Then $\{ X_{ij} \to X\} _{i \in I, j \in J_ i}$ is a family as in Definition 87.11.1 for $X$. $\square$
Lemma 87.15.2. 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 87.11.1. Then $\{ X_ i \times _ Z Y_ j \to X \times _ Z Y\} $ is a family of maps which are representable by algebraic spaces and étale. Thus Lemma 87.15.1 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 87.9.1. 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 87.13.1. $\square$
Lemma 87.15.3. Let $S$ be a scheme. The category of formal algebraic spaces over $S$ has fibre products.
Proof. Special case of Lemma 87.15.2 because formal algebraic spaces have representable diagonals, see Lemma 87.11.2. $\square$
Lemma 87.15.4. 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 87.12.1. $\square$
We have already proved the following lemma (without knowing that fibre products exist).
Lemma 87.15.5. 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 87.11.2. $\square$
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