The Stacks project

Lemma 29.44.3. Let $f : X \to S$ be a morphism of schemes. The following are equivalent:

  1. The morphism $f$ is finite.

  2. There exists an affine open covering $S = \bigcup U_ i$ such that each $f^{-1}(U_ i)$ is affine and $\mathcal{O}_ S(U_ i) \to \mathcal{O}_ X(f^{-1}(U_ i))$ is finite.

  3. There exists an open covering $S = \bigcup U_ i$ such that each $f^{-1}(U_ i) \to U_ i$ is finite.

Moreover, if $f$ is finite then for every open subscheme $U \subset S$ the morphism $f : f^{-1}(U) \to U$ is finite.

Proof. See Algebra, Lemma 10.36.14. Some details omitted. $\square$

Comments (1)

Comment #211 by Rex on

Typo: "subscheme U⊂S the morphim"

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