etale neighborhoods

Question

I've read that quasi-compact etale morphisms of schemes over a not necessarily algebraiclly closed field $F$ (I'm happy to take $F$ a field of char $0$) are the algebraic analogs of local diffeomorphisms. I also know that etale morphisms are open maps, but not usually open immersions (for this, we also need "radiciality"). Now, my question concerns the notion of etale neighborhoods. Let $f : Y \to X$ be a quasi-compact etale map of $F$-schemes (I'm happy to take $X,Y$ to be smooth and proper and $X$ also geometrically connected). Let $y \in Y$ and let $x = f(y) \in X$ and let $\phi: (S,y) \to (X,x)$ be an etale neighborhood. My intuition tells me that, since the word "neighborhood" is there, I should be able to make $S$ small enough (but still an etale neighborhood) such that $\phi$ is an open immersion. Is this intuition correct? If not, how can we get something resembling a local isomorphism between $S$ and $X$?

Answer 1

Etale morphisms are not open immersions locally in the Zariski topology. This is the reason for introducing the etale topology.

To say a little more: etale morphisms can be characterized as those morphism which induce isomorphisms on tangent spaces (say if the morphism is between smooth varieties over an alg. closed field), and one knows that, in the world of complex manifolds, such morphisms are locally (on the source) open immersions (and this is an important and useful fact!). One would like to have an analogue in the world of schemes, and so one defines a new topology in which etale morphisms are declared to be "open immersions"; more precisely, they define the covers in the etale site.

In usual topology, we intersect with a n.h. of a point, or pull-back a n.h. of a point under a morphism, to study various phenomena locally at the point in question. If you remember that intersecting with an open subset is the same as fibre producing with the corresponding open immersion, then you can see that the analogous process of investigating phenomena locally in the etale topology is carried out by pulling-back with respect to etale morphisms.

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E.g. if $f: Y \to X$ is etale, then the diagonal $Y \to Y\times_X Y$ is an open immersion (more generally, this is the hallmark of unramified morphisms).

If we compose this with either projection $Y \times_X Y \to Y$ we get the identity. Thus we may find an open subset (namely $\Delta(Y)$) of $Y \times_X Y$ such that (either of) the projection(s) $Y\times_X Y \to Y$ restricted to this open subset becomes the identity (and in particular an open immersion).

This gives some meaning to the statement that an etale morphism is an open immersion stale locally.

(If $Y \to X$ is furthermore finite and Galois, then $Y\times_X Y \to Y$ actually decomposes as a disjoint union of copies of $Y$, and (either) projection, when restricted to one of these copies of $Y$, is an isomorphism. Thus in this case, the map $f$, etale locally, becomes an open immersion in a n.h. of every point in its source.)

[Note: the above is slightly edited from its original form in an attempt to repair nonsense, as pointed out in comments. Rather than write to much more in this salvage attempt, it might be better just to say that there is a slogan: etale locally, unramified morphisms are immersions, which is made precise by this result. If we apply this to etale morphisms, the corresponding statement becomes etale locally, etale morphisms are open immersions, which is what I was trying to get at.]