Etale maps are covering maps I am looking for a reference for the following fact (I hope it is a fact, if not, can someone give a more precise formulation)?
Suppose $V$ and $W$ are two algebraic varieties over an algebraically closed field. Then a morphism $f:V\to W$ is an etale map if and only if, the induced map on each Zariski cotangent space is an isomorphism.
By an etale map I mean a flat unramified map of finite type.
Thank you!
 A: I think you need to assume that $V$ and $W$ are smooth varieties; then you'll find the statement (or, at least, a very similar one about stalks) in SGA 1, II, Corollaire 4.6 or (in English) Bosch, Lütkebohmert, Raynaud, Néron Models, 2.2, Proposition 10.
I've been trying to come up with a good counterexample with non-smooth varieties.  Perhaps you can do something like this: take $W = l_1 \cup l_2$ to be the union of two lines in the plane, and $V = l'_1 \cup l'_2 \cup l'_3$ the union of three lines in the plane meeting in a point.  They both have one singular point, at which the tangent space has dimension 2.  Map $V \to W$ by sending $l'_1$ to $l_1$, and both $l'_2$ and $l'_3$ to $l_2$, so that the singular point of $V$ maps to the singular point of $W$.  Then I believe you get an isomorphism of tangent spaces at each point, but the map isn't étale.
A: I believe the correct version of this is Proposition I.2.9 in Milne's Lectures on Etale Cohomology. (I see that there is already a reference given, but this one is available online freely.) 
