Everything you've ever wanted to know about Etale Cohomlogy but were afraid to ask

This is going to be a short post to attempt to explain the 'point' of etale cohomology and the etale fundamental group. The latter is often called the algerbaic fundamental group.

In case you don't know what any of these things are, they are important in the study of arithmetic algebraic geometry. It is an area of mathematics somewhere between algebraic geometry and number theory. Lang called it Diophantine geometry. If you don't know what those words mean, what follows will not make sense to you.

To begin with, the most readable introduction to this area across which I have come is in Milne's online notes. They are incomplete and lack many proofs, but they give one the general idea. I don't aim here to teach anyone about etale cohomolgy or the fundamental group, but if you are learning about them and you happen to stumble across this site (a concurrence which is mind-bogglingly unlikely), then this may help you to see the big picture.

First of all, start by understanding the fundamental group. Consider two cases. If our base space is a field, the fundamental group is its absolute Galois group. If the base is a complex variety, the etale fundamental group is closely related to its regular fundamental group. For any other scheme, it is different, but you should understand that it generalises these very different notions, one purely geometric and the other purely algebraic.

Its importance in the study of etale cohomolgy lies in the following statement: Connected etale covers of a scheme correspond to transitive sets acted on by the etale fundamental group. The analogy with the geometric case is perfect, because connected covering spaces of a nice (if these comments are making sense to you so far, you'll know what nice means) space correspond to transitive sets acted on by the regular fundamental group.

Then the basic construction of the machinery of etale cohomology is very similar to that of regular sheaf cohomology, but extra algebraic input is often need for standard proofs.

However, the cohomology groups produced by etale cohomology are very different from those produced from Serre-style quasi coherent chomolgy theory. The real value of etale cohomology lies in these differences. They are sufficiently many and large to require little comment.

It is often said that etale cohomology (especially in characteristic zero) has two main ingredients: Galois cohomology and clasical topology. This is made most concrete by comparing the cohomlogy of a k-scheme to that of its base change to the algebraic closure of k. The relationship is broken down by Grothendieck's spectral sequence theorem into precisely the algebraic and geometric data for which one would hope.

If you make it this far, please leave a comment. I'd love to discuss this with someone.