The proof of the Weil conjectures is one of these great achievements of modern mathematics which has more trouble than it should diffusing out of its original field. It’s understandable why it would be scary to people. It involves things over characteristic , which is a pretty scary place, and a lot of hard algebraic geometry.
On the other hand, I think it’s a shame that so many people who like to think about categorification don’t realize arithmetic geometers have been doing it for decades; thus, I’m going to try to write some posts which translate.
The important fact here is the function-sheaf correspondence. In this post, I’ll just try to tell you what kind of sheaves we’ll have, and then in a forthcoming one, we’ll talk about the functions (surprisingly, yes, it makes sense to talk about the sheaves first).
Now, the kind of sheaves we’d like to work with are “constructible” ones. Let me explain these for a variety over first. The most important example of a constructible sheaf is the sheaf of locally constant functions on a variety X (Exercise: why locally constant? Why aren’t constant functions a sheaf?).
There are two important ways to generalize this:
- We could instead take a vector bundle, and look at the sheaf of flat sections of that (locally constant functions are looking at the trivial vector bundle with the obvious connection). Such a sheaf is called a local system.
- We could instead take a locally closed subvariety and look at the sheaf that assigns to the space of locally constant functions (or flat sections of a bundle) on .
Why are these sheaves interesting? Well, over , if you look at the DeRham complex on X (or the complex of singular cochains) as a complex of sheaves, what will you get? On each small enough open set, the cohomology of the DeRham complex on that set is just , the constant functions on that set. We have just proved:
Proposition. The sheaf of locally constant functions is quasi-isomorphic to the DeRham complex (cochain complex) as sheaves. In fact, the sheaf cohomology is the same as the DeRham cohomology (singular cohomology).
So these sheaves are a local way to think about cohomology. But we had to something non-algebraic; when I said “small enough open set” I was thinking of $X$ as a complex manifold, not a variety. At some point, the genius idea appeared of trying to model the complex topology on a complex variety as something algebraic using the “etale topology” which is not a topology at all, but rather a clever generalization of a topology called a site. I don’t want to get into the details of this for now, so let’s treat it as a black box.
This was mainly intended to convince you that when I look at a variety X over , and I talk about the sheaf on this variety, I am not just making stuff up; I have precise mathematics in mind, even if I haven’t explained it fully.
So, I want a category of sheaves: for now let’s start with X, and cut it into pieces (for example, imagine that X has a group G acting on it, and these are the orbits), and look at the category of complexes of sheaves such that the restriction to of each degree of cohomology of the complex (I’m not doing any sheaf cohomology here, just playing with chain complexes) is a local system. We’ll call this the derived category of -contructible sheaves on , though this is a little bit of an abuse of terminology.
Of course, all the cool kids will think of this as a derived category, by inverting quasi-isomorphisms (and the really cool kids will think it’s actually an category), but that’s not super important for our purposes.
So, an example of an object in this category would be a local system. Or a local system on one of the ‘s, thought of as a sheaf on the whole thing, as I described for the constant sheaf. Or some weird way of gluing a bunch of these together.
Passing to a finite field
Now, we want to start reducing mod q, where q is a prime power (there’s no harm in thinking it’s just a prime). I know that thinking about varieties over is a bit hard for geometrically/topologically minded people, but the amazing thing is, they really aren’t that different from varieties over , in a way that we’ll make precise shortly.
So assume that everything you did (including the definition of the ) involved polynomials with integer coefficients (there are ways to fix things if they’re just algebraic, but that’s getting way too technical for us). Then one has a variety over and one over , given by reducing all of those polynomials mod p, along with pieces and . (Note: I’m using the algebraic closure now).
Interestingly, one can makes sense of the idea of a local system on a variety over , using this fancy etale stuff. Though, it’s important to note, my sheaves are still of vector spaces over ; what I’ve changed is just the variety.
Then one has a miraculous theorem, which the biggest names behind are Grothendieck and Deligne (though I read it in the novella of Beilinson, Bernstein and Deligne).
Theorem. Choose your favorite finite collection of local systems on (for example, you should just take the constant sheaf ). These have extensions by base change to and
The derived categories of -contructible sheaves that use only these local systems on in the complex topology and in the etale topology are canonically* equivalent.
*Health warning: As usual in blog posts (at least those written by me, and not Terry Tao), you should should look more for the spirit of the theorem than the details. One subtlety I’ve quite knowingly thrown to the wind is that I’ve secretly picked an isomorphism between and . The fact that this was a non-canonical choice has interesting consequences, but puts a serious caveat in front of my “canonical” here. I’m sure I’ve skipped over some others, but bear with me.
In particular, since the cohomology of your variety is just , this is the same whether you compute it over or (remember, when I say compute over, I mean the coefficients of your variety. Of course, if you change the field in the coefficients of your sheaf, then everything can change, as any algebraic topologist will tell you). Incroyable!
In our next installment, I’ll explain what all of this has to do with categorification.
EDIT: I’m very far from being an arithmetic geometer (or even a proper algebraic geometer) and learned this material in somewhat piecemeal fashion, so am happy to accept historical notes or corrections. Matthew Emerton notes that the names of Grothendieck and Michael Artin also deserve to be mentioned in connection with the theorem above.