Lemma 80.3.9 (046D)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 80: Bootstrap
Section 80.3: Morphisms representable by algebraic spaces
Lemma 80.3.9 (cite)

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Lemma 80.3.9. Let $S$ be a scheme. Let $F_ i, G_ i : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ , $i = 1, 2$ . Let $a_ i : F_ i \to G_ i$ , $i = 1, 2$ be representable by algebraic spaces. Then

$a_1 \times a_2 : F_1 \times F_2 \longrightarrow G_1 \times G_2$

is a representable by algebraic spaces.

Proof. Write $a_1 \times a_2$ as the composition $F_1 \times F_2 \to G_1 \times F_2 \to G_1 \times G_2$ . The first arrow is the base change of $a_1$ by the map $G_1 \times F_2 \to G_1$ , and the second arrow is the base change of $a_2$ by the map $G_1 \times G_2 \to G_2$ . Hence this lemma is a formal consequence of Lemmas 80.3.8 and 80.3.3. $\square$

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