The Stacks project

Lemma 37.64.8. Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes. Then the pushout $Y \amalg _ Z X$ exists in the category of schemes. Picture

\[ \xymatrix{ Z \ar[r]_ i \ar[d]_ j & X \ar[d]^ a \\ Y \ar[r]^-b & Y \amalg _ Z X } \]

The diagram is a fibre square, the morphisms $a$ and $b$ are closed immersions, and there is a short exact sequence

\[ 0 \to \mathcal{O}_{Y \amalg _ Z X} \to a_*\mathcal{O}_ X \oplus b_*\mathcal{O}_ Y \to c_*\mathcal{O}_ Z \to 0 \]

where $c = a \circ i = b \circ j$.

Proof. This is a special case of Proposition 37.64.3. Observe that hypothesis (3) in Situation 37.64.1 is immediate because the fibres of $j$ are singletons. Finally, reverse the roles of the arrows to conclude that both $a$ and $b$ are closed immersions. $\square$

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