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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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