The Stacks project

Lemma 67.46.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $f$ is finite locally free,

  2. $f$ is finite, flat, and locally of finite presentation.

If $Y$ is locally Noetherian these are also equivalent to

  1. $f$ is finite and flat.

Proof. In each of the three cases the morphism is representable and you can check the property after base change by a surjective étale morphism $V \to Y$, see Lemmas 67.45.3, 67.46.3, 67.30.5, and 67.28.4. If $Y$ is locally Noetherian, then $V$ is locally Noetherian. Hence the result follows from the corresponding result in the schemes case, see Morphisms, Lemma 29.48.2. $\square$

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