The Stacks project

Lemma 40.9.1. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $g : U' \to U$ be a morphism of schemes. Let $(U', R', s', t', c')$ be the restriction of $(U, R, s, t, c)$ via $g$.

  1. If $s, t$ are locally of finite type and $g$ is locally of finite type, then $s', t'$ are locally of finite type.

  2. If $s, t$ are locally of finite presentation and $g$ is locally of finite presentation, then $s', t'$ are locally of finite presentation.

  3. If $s, t$ are flat and $g$ is flat, then $s', t'$ are flat.

  4. Add more here.

Proof. The property of being locally of finite type is stable under composition and arbitrary base change, see Morphisms, Lemmas 29.15.3 and 29.15.4. Hence (1) is clear from Diagram (40.9.0.1). For the other cases, see Morphisms, Lemmas 29.21.3, 29.21.4, 29.25.6, and 29.25.8. $\square$

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