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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