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 $S$.

Elementary distinguished squares were defined in Derived Categories of Spaces, Section 75.9.

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. $\square$

Lemma 81.9.2. Let $S$ be a scheme. Let $V$, $U$ be algebraic spaces over $S$. Let $V' \subset V$ be an open subspace and let $f' : V' \to U$ be a separated étale morphism of algebraic spaces over $S$. Then there exists a pushout

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

in the category of algebraic spaces over $S$ and moreover $(U \subset W, f : V \to W)$ is an elementary distinguished square.

**Proof.**
We are going to construct $W$ as the quotient of an étale equivalence relation $R$ on $U \amalg V$. Such a quotient is an algebraic space for example by Bootstrap, Theorem 80.10.1. Moreover, the proof of Lemma 81.9.1 tells us to take

\[ R = U \amalg V' \amalg V' \amalg V \amalg (V' \times _ U V' \setminus \Delta _{V'/U}(V')) \]

Since we assumed $V' \to U$ is separated, the image of $\Delta _{V'/U}$ is closed and hence the complement is an open subspace. The morphism $j : R \to (U \amalg V) \times _ S (U \amalg V)$ is given by

\[ u,\ v',\ v',\ v,\ (v'_1, v'_2) \mapsto (u, u),\ (f'(v'), v'),\ (v', f'(v')),\ (v, v),\ (v'_1, v'_2) \]

with obvious notation. It is immediately verified that this is a monomorphism, an equivalence relation, and that the induced morphisms $s, t : R \to U \amalg V$ are étale. Let $W = (U \amalg V)/R$ be the quotient algebraic space. We obtain a commutative diagram as in the statement of the lemma. To finish the proof it suffices to show that this diagram is an elementary distinguished square, since then Lemma 81.9.1 implies that it is a pushout. Thus we have to show that $U \to W$ is open and that $f$ is étale and is an isomorphism over $W \setminus U$. This follows from the choice of $R$; we omit the details. $\square$

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