References for quasi finite and proper implies finite

Question

Does anybody know a reference for a proof of:

Let $f: X \rightarrow Y$ be a quasi-finite proper morphism of varieties. Then $f$ is finite.

Is there one in Hartshorne? I could not find it.

Thanks!

Answer 1

The proof is in ÉGA IV 3. It is in two steps.

Step I : reduce to the case of $\mathrm Y = \mathrm{Spec}(\mathrm A)$ where $\mathrm A$ is a complete noetherian local ring.

Step II : quasi-finite $\mathfrak m_{\mathrm A}$-seperated modules are finite for such an $\mathrm A$.

Answer 2

I think this is just a generalization of the fact that finite integral extensions of domains are finitely generated module over the lower integral domain.

Anyway if you need a reference/proof to the statement in your post, you can cite Lemma 02OG of the stacks project.