The Stacks project

80.7 Pushouts along closed immersions and integral morphisms

This section is analogue of More on Morphisms, Section 37.65.

Lemma 80.7.1. In More on Morphisms, Situation 37.65.1 let $Y \amalg _ Z X$ be the pushout in the category of schemes (More on Morphisms, Proposition 37.65.3). Then $Y \amalg _ Z X$ is also a pushout in the category of algebraic spaces over $S$.

Proof. This is a consequence of Lemma 80.3.1, the proposition mentioned in the lemma and More on Morphisms, Lemmas 37.65.6 and 37.65.7. Conditions (1) and (2) of Lemma 80.3.1 follow immediately. To see (3) and (4) note that an ├ętale morphism is locally quasi-finite and use that the equivalence of categories of More on Morphisms, Lemma 37.65.7 is constructed using the pushout construction of More on Morphisms, Lemmas 37.65.6. Minor details omitted. $\square$


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