Quasi-finite + separated but not finite morphism

What is an interesting example of that? Things like $Spec(K) \to Spec(L)$ do not count cause they are not interesting.

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An open embedding that is not closed. I don't how "interesting" that is. – Angelo Sep 24 '10 at 4:54
Dear Dung: Are you aware of Zariski's Main Theorem, which in one form says that Angelo's example and your example are nearly the only examples? To be precise, a quasi-finite separated map $X \rightarrow S$ with $S$ quasi-compact and quasi-separated (e.g., noetherian) always factors as an open immersion into a finite $S$-scheme. So by the transitive property of "not interesting", can we say that all quasi-finite separated maps are "not interesting" (at least locally on the base)? I'd hope not, because I can think of interesting ones (e.g., torsion in semi-abelian schemes). – BCnrd Sep 24 '10 at 5:37
I do not understand, what this $\operatorname{Spec}(K)\to\operatorname{Spec}(L)$ example should be. I assume, that $L$ is a field and $K/L$ a field extension. But then we do have: If $\operatorname{Spec}(K)\to\operatorname{Spec}(L)$ is quasi-finite, then $K/L$ is a finite extension and $\operatorname{Spec}(K) \to \operatorname{Spec}(L)$ is a finite morphism. Right? – Sebastian Petersen Sep 28 '10 at 7:30
Sorry, something went wrong above. I cannot delete / edit the comment. I do not understand, what this $Spec(K)\to Spec(L)$ example should be. I assume $L$ is a field and $K/L$ a field extension. But then we do have: If $Spec(K)\to Spec(L)$ is quasi-finite, then $K/L$ is finite and $Spec(K)\to Spec(L)$ is a finite morphism. Right? – Sebastian Petersen Sep 28 '10 at 7:33
It does not have to be a finite extension. – John Doe Oct 9 '10 at 2:44

Dear Dung, a pleasantly geometric example of a quasi-finite, separated, but not finite morphism is the projection of the hyperbola $xy=1$ in the affine $x,y$ plane on the $x$-axis. Its image is the affine line minus the origin. It is clearly quasi-finite (even injective) but not finite, since its image is not closed . (Also, morphisms between affine schemes are separated)