The Stacks project

Lemma 29.44.11. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. $f$ is finite, and

  2. $f$ is affine and proper.

Proof. This follows formally from Lemma 29.44.7, the fact that a finite morphism is integral and separated, the fact that a proper morphism is the same thing as a finite type, separated, universally closed morphism, and the fact that an integral morphism of finite type is finite (Lemma 29.44.4). $\square$

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