The Stacks project

Lemma 37.67.9. Let $i : Z \to X$ and $j : Z \to Y$ be closed immersions of schemes. Let $f : X' \to X$ and $g : Y' \to Y$ be morphisms of schemes and let $\varphi : X' \times _{X, i} Z \to Y' \times _{Y, j} Z$ be an isomorphism of schemes over $Z$. Consider the morphism

\[ h : X' \amalg _{X' \times _{X, i} Z, \varphi } Y' \longrightarrow X \amalg _ Z Y \]

Then we have

  1. $h$ is locally of finite type if and only if $f$ and $g$ are locally of finite type,

  2. $h$ is flat if and only if $f$ and $g$ are flat,

  3. $h$ is flat and locally of finite presentation if and only if $f$ and $g$ are flat and locally of finite presentation,

  4. $h$ is smooth if and only if $f$ and $g$ are smooth,

  5. $h$ is étale if and only if $f$ and $g$ are étale, and

  6. add more here as needed.

Proof. We know that the pushouts exist by Lemma 37.67.8. In particular we get the morphism $h$. Hence we may replace all schemes in sight by affine schemes. In this case the assertions of the lemma are equivalent to the corresponding assertions of More on Algebra, Lemma 15.7.7. $\square$

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