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.