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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.


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Your question is only the "second thing" in the title. –  user18119 Jan 29 '13 at 15:47
An overkill proof via ZMT: By the Grothendieck version of Zariski's main theorem, $f$ is the composition of an open embedding $g: X\rightarrow Z$ and a finite morphism $h:Z\rightarrow Y$. Then, $g$ is the composition of the graph morphism $X \rightarrow X\times_Y Z$ and the projection $X\times_Y Z \rightarrow Z.$ The first is a base-change of the morphism $Z \rightarrow Z\times_Y Z$, which is a closed embedding and thus proper, and the second is a base change of $f$, and thus proper. Therefore, $g$ is proper and thus an isomorphism (as the image must be both closed and open.) –  Nehsb Jan 29 '13 at 17:29
@QiL, oops, found a reference for the first thing while typing and forgot to change the title, thanks. –  Joachim Jan 29 '13 at 17:36
@Nehsb, why don't you post this as an answer? –  Joachim Jan 29 '13 at 17:37
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1 Answer

up vote 1 down vote accepted

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 $\mathrm A$.

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