# Why is this map almost an étale covering?

Let $V$ and $W$ be absolutely irreducible quasi-projective varieties over $K$ with char $K=0$ with dim $V=$ dim $W$ and let $f: W \rightarrow V$ be a dominant morphism. Why is it possible to make $f$ into an étale covering by restricting $V$ ?
I know that this is probably a very basic and silly question but I have only started reading about algebraic geometry.

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Generically, the map is etale. That is, there exists a nonempty open set $U \subset V$ such that $f^{-1}(U) \to U$ is finite and etale. Indeed, we can assume $f$ is flat at the outset by the generic flatness theorem. Then this follows because if $\xi$ is the generic point of $V$, $f^{-1}(\xi) \to \xi$ is a finite etale morphism (it's the extension of function fields, which is finite and separable as we are in characteristic zero).
Alternatively, the sheaf $\Omega_{W/V}$ is zero at the generic point of $W$ for the same reasons above, so its support is a proper closed subset $Z \subset W$. If we take $U = V- \overline{f(Z)}$, then the map $f^{-1}(U) \to U$ is unramified (and as above we can shrink further, if necessary, to get flatness).
Finally, to get finiteness, again we observe that $f^{-1}(\xi) \to \xi$ is finite, and since $\xi$ is the inverse limit over all nonempty open sets $U \subset V$, it follows that $f^{-1}(U) \to U$ is finite for some $U$. (This is a special case of the "noetherian descent" argument expounded at length in EGA IV-8.)
Could you please explain what exactly you meant by "Then this follows because if $\xi$ is the generic point of $V$, $f^{-1}(\xi) \to \xi$ is a finite etale morphism"? –  kalb Oct 11 '11 at 18:45
@kalb: In general, there are a bunch of results of the following kind: if $f: X \to Y$ is a morphism of finite presentation and $f^{-1}(\xi} \to \xi$ has some property (where $\xi$ is the generic point), then $f^{-1}(U) \to U$ has that property for $U$ an open neighborhood of $\xi$. One way to see this is to use the general formalism ("noetherian descent") that if $f_\alpha: X_\alpha \to Y_\alpha$ is a morphism of inverse systems of schemes, with the transition maps affine, then if $f: \varprojlim X_\alpha \to \varprojlim Y_\alpha$ has some property, then so does $f_\alpha$ –  Akhil Mathew Oct 12 '11 at 3:14
...for $\alpha$ sufficiently large. In the case here, the point is that $\xi$ is the inverse limit of the open affine sets in $Y$. –  Akhil Mathew Oct 12 '11 at 3:15