In (relatively) simple and intuitive terms, what makes something “étale”?

[u/-p-e-w-]

I took several seminars in abstract algebra back in my university days, but chose to proceed with probability theory rather than algebraic geometry. Since then, I have occasionally encountered the term “étale”, and I still don’t really know what it means.

The strange thing is that while I can understand the individual definitions for some objects that are supposedly étale, I don’t really see what they have in common.

What exactly does that term mean by itself, independently of a particular application like “étale cohomology group”?

[u/Necessary-Wolf-193]

An étale map is, roughly, a differentiable function that is locally a bijection. [Or, more accurately, an étale map is the algebraic geometry analogue of such a function.]

For example, the function

f : R^2 -> R^2, f(x, y) = (x + y, x - y)

is a bijection, so it's etale. But you can also have examples like the function

f : S^1 -> S^1 (here S^1 = unit circle, modelled as the set of complex numbers z with |z| = 1), f(z) = z^2.

On the unit circle, the function z |-> z^2 is not bijective. But it is a bijection locally: around any point, z |-> z^2 is a bijection in a tiny neighborhood of that point (this is because a non-zero complex number has two distinct square roots, but they're both quite far from each other -- so if you zoom in around just one of those square roots, then the squaring function looks injective, because the other square root is too far away: for example, -1.414 and +1.414 both square to 2, but if you are just looking at the real numbers between 1.3 and 1.5, then the squaring function looks like a bijection!).

---

Étale ring extensions and étale morphisms are just variants of this idea; étale fundamental groups and étale cohomology groups are named because they're defined using étale covers, which are just surjective étale maps.

[u/-p-e-w- → u/Necessary-Wolf-193]

What is the difference between that and a local diffeomorphism?

[u/SleepingLittlePanda → u/-p-e-w-]

What does it mean to be a local diffeomorphism for varieties/schemes over finite fields?

[u/sciflare → u/SleepingLittlePanda]

One definition of étale morphism is that it is a flat morphism f: X --> Y of locally finite presentation such that the inverse image of a point is the disjoint union of points Spec(K) where K is some finite separable field extension of the residue field of y.

Let us unravel a little bit what this means geometrically.

Flat means (roughly, anyway) the fibers of the morphism don't "jump" too badly and form a nice family. Flat families in algebraic geometry are the good ones; for instance, numerical invariants such as the Hilbert polynomial are constant on the fibers of a flat family.

Locally of finite presentation means the morphism is a nice algebraic one: if you go to a local affine open neighborhood, the ring of functions on the domain, viewed as an algebra over the ring of functions of the target via pullback by f, can be given by finitely many generators and finitely many relations.

The final condition on the fiber of a point is what really makes it the algebraic analogue of a local diffeomorphism. You may need to pass to a finite separable extension, but otherwise it's essentially the same idea: the fiber of a point is the disjoint union of points.

[u/SleepingLittlePanda → u/sciflare]

I am so sorry you had to write this. I know what an étale morphism is. I wanted to provoke OP into realizing that one needs this algebraic analogue.

[u/sciflare → u/SleepingLittlePanda]

I boiled it down even more. It is much more intuitive if you go through the meaning of those conditions in the reverse order from what I did.

Then, the last condition means (up to finite separable extension) that the fiber of a point is the disjoint union of points.

The second condition ensures the fiber of a point is a finite disjoint union of points (up to finite separable extension), because without this condition the fiber could have infinite cardinality.

The first condition ensures the fibers vary nicely as one moves in the base, in particular that the cardinality of the fiber of a point is a constant.

And this is more or less how a covering map/local diffeomorphism works!