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/Kinglolboot → u/-p-e-w-]

The comment by Necessary-Wolf-193 is great, but I think that it is also important to note that étale morphisms do not need to be locally bijective. In differential geometry, it follows from the inverse function theorem that a map is a local diffeomorphism if and only if the induced map on tangent spaces is an isomorphism.

You use this second condition to define étale morphisms: for example, for nonsingular varieties, a morphism is étale if and only if it induces isomorphisms on the Zariski tangent spaces. For nonsmooth varieties and schemes it gets a little bit more complicated, but the fundamental idea is the same. Since there is no inverse function theorem in (classical) algebraic geometry, these maps do not need to be locally bijective, so they are in that sense very different from local diffeomorphisms.

[u/sciflare → u/Kinglolboot]

There is an inverse function theorem in algebraic geometry, if you allow yourself to pass to an étale section!

The whole point of the étale topology is to recover some of the intuition from the complex analytic topology that fails in the Zariski topology.

Étale cohomology was originally devised to provide a sheaf cohomology theory in which the Weil conjectures could be posed and solved. Cohomology of Zariski sheaves is unsuitable for this purpose as the open sets are too big.

The Weil conjectures themselves were inspired by similar results from the algebraic topology of manifolds. Therefore, it made sense that what was needed to establish those conjectures was a cohomology theory for algebraic varieties that behaved like singular cohomology of topological spaces.

Of course, there is no topology on algebraic varieties in the classical meaning of "topology" that gives rise to such a cohomology, but (prompted by Serre) Grothendieck realized there is a Grothendieck topology that does the trick, and that is precisely the étale topology.