Étale cohomology

Okay, so imagine you have a bunch of shapes and you want to understand something about them. Maybe you want to know how many corners they have, or how many sides they have.

Now, étale cohomology is kind of like that, but instead of shapes, we're looking at something called a "scheme." A scheme is a lot like a shape, but it's a mathematical object that we study in algebraic geometry.

And just like with shapes, we want to understand different things about schemes. In particular, we want to understand something called the "cohomology" of the scheme.

Now, cohomology is kind of like a way of measuring the "holes" in the scheme. So if you have a shape with a hole in it, its cohomology will be different from a shape without a hole.

But here's the thing: in algebraic geometry, we can't just count corners and sides like we can with shapes. Instead, we have to use something called "sheaf cohomology," which is a fancy way of calculating the cohomology of the scheme.

And that's where étale cohomology comes in. It's a special kind of sheaf cohomology that works well for certain kinds of schemes. With étale cohomology, we can study the holes in the scheme in a really powerful way, and use that to understand a lot about the scheme and its properties.

So in the end, étale cohomology is really just a way of studying the "holes" in schemes, kind of like how we study the corners and sides of shapes. But because algebraic geometry is such a complex subject, we have to use some pretty fancy tools (like sheaf cohomology) to do it!