The Stacks project

Lemma 37.14.6. In the situation of Lemma 37.14.4. If $V' = G(V, U', \varphi )$ for some triple $(V, U', \varphi )$, then

  1. $V' \to Y'$ is locally of finite type if and only if $V \to Y$ and $U' \to X'$ are locally of finite type,

  2. $V' \to Y'$ is flat if and only if $V \to Y$ and $U' \to X'$ are flat,

  3. $V' \to Y'$ is flat and locally of finite presentation if and only if $V \to Y$ and $U' \to X'$ are flat and locally of finite presentation,

  4. $V' \to Y'$ is smooth if and only if $V \to Y$ and $U' \to X'$ are smooth,

  5. $V' \to Y'$ is étale if and only if $V \to Y$ and $U' \to X'$ are étale, and

  6. add more here as needed.

If $W'$ is flat over $Y'$, then the adjunction mapping $G(F(W')) \to W'$ is an isomorphism. Hence $F$ and $G$ define mutually quasi-inverse functors between the category of schemes flat over $Y'$ and the category of triples $(V, U', \varphi )$ with $V \to Y$ and $U' \to X'$ flat.

Proof. Looking over affine pieces the assertions of this lemma are equivalent to the corresponding assertions of More on Algebra, Lemma 15.7.7. $\square$

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