The Stacks project

Lemma 37.61.7. In Situation 37.61.1 the category of schemes flat, separated, and locally quasi-finite over the pushout $Y \amalg _ Z X$ is equivalent to the category of $(X', Y', Z', i', j', f, g, h)$ as in Lemma 37.61.6 with $f, g, h$ flat. Similarly with “flat” replaced with “étale”.

Proof. If we start with $(X', Y', Z', i', j', f, g, h)$ as in Lemma 37.61.6 with $f, g, h$ flat or étale, then $Y' \amalg _{Z'} X' \to Y \amalg _ Z X$ is flat or étale by More on Algebra, Lemma 15.7.7.

For the converse, let $W \to Y \amalg _ Z X$ be a separated and locally quasi-finite morphism. Set $X' = W \times _{Y \amalg _ Z X} X$, $Y' = W \times _{Y \amalg _ Z X} Y$, and $Z' = W \times _{Y \amalg _ Z X} Z$ with obvious morphisms $i', j', f, g, h$. Form the pushout $Y' \amalg _{Z'} X'$. We obtain a morphism

\[ Y' \amalg _{Z'} X' \longrightarrow W \]

of schemes over $Y \amalg _ X Z$ by the universal property of the pushout. If we do not assume that $W \to Y \amalg _ Z X$ is flat, then in general this morphism won't be an isomorphism. (In fact, More on Algebra, Lemma 15.6.5 shows the displayed arrow is a closed immersion but not an isomorphism in general.) However, if $W \to Y \times _ Z X$ is flat, then it is an isomorphism by More on Algebra, Lemma 15.7.7. $\square$


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