Lemma 81.9.1. Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary distinguished square. Then

is a pushout in the category of algebraic spaces over $S$.

Lemma 81.9.1. Let $S$ be a scheme. Let $(U \subset W, f : V \to W)$ be an elementary distinguished square. Then

\[ \xymatrix{ U \times _ W V \ar[r] \ar[d] & V \ar[d]^ f \\ U \ar[r] & W } \]

is a pushout in the category of algebraic spaces over $S$.

**Proof.**
Observe that $U \amalg V \to W$ is a surjective étale morphism. The fibre product

\[ (U \amalg V) \times _ W (U \amalg V) \]

is the disjoint union of four pieces, namely $U = U \times _ W U$, $U \times _ W V$, $V \times _ W U$, and $V \times _ W V$. There is a surjective étale morphism

\[ V \amalg (U \times _ W V) \times _ U (U \times _ W V) \longrightarrow V \times _ W V \]

because $f$ induces an isomorphism over $W \setminus U$ (part of the definition of being an elementary distinguished square). Let $B$ be an algebraic space over $S$ and let $g : V \to B$ and $h : U \to B$ be morphisms over $S$ which agree after restricting to $U \times _ W V$. Then the description of $(U \amalg V) \times _ W (U \amalg V)$ given above shows that $h \amalg g : U \amalg V \to B$ equalizes the two projections. Since $B$ is a sheaf for the étale topology we obtain a unique factorization of $h \amalg g$ through $W$ as desired. $\square$

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