So, I started this series a while back with this post, by talking about some sheaves on an algebraic variety. This post will probably not make a lick of sense to those who haven’t read the first part yet, though you’re welcome to try. The upshot of that was that one has a machine into which one can can feed in
- a variety defined over (i.e. defined by polynomials with integer coefficients)
- a choice of stratification of (a good example might be the orbits of an algebraic group with finitely many orbits, so for example, the Schubert cells on a Grassmannian)
- a set of acceptable local systems valued in your favorite field k (though, if want to do things properly, your favorite field should probably be ).
and receive out the other end a category. The cool fact about this category is that you can think of it as sheaves on the complex variety or its analogue over characteristic p (let p be your favorite prime) and you will get the same answer. The former category has the advantage of involving geometry that you probably care about, like the cohomology of smooth varieties, and the latter has the advantage that there is a Frobenius acting.
What I’d like to explain in this post how to analyze the structure of this category, and what that has to do with categorification. This will require a bit of machinery, but believe me, the result will justify it.
If you want to categorify things, one thing you’d like to have around is a bunch of categories whose Grothendieck groups you understand. That way, if you’re in the market for a categorification, you know where to browse.
Now, what we would like to do is analyze complexes of constructible sheaves by associating functions to them, in a way that gives information about the Grothendieck group. The functions we would really like to use are the Poincaré polynomials (graded dimensions of the cohomology, and I mean as a complex, not sheaf cohomology) of their stalks. This would send, for example, the constant local system to the constant 1 function, which is good. The problem is, this would send every local system to a constant function, which is bad. Even worse, the Poincaré polynomial isn’t an invariant of Grothendieck classes, because there’s a long exact sequence on stalks for each exact sequence in the category of complexes of sheaves. Instead, all you get to keep is the Euler characteristic, which is lovely, but for our purposes pretty weak tea.
However, the Euler characteristic can point us in the right direction. What is it? If one thinks of the dimension of a vector space as the trace of its identity operator, then the Euler characteristic is the supertrace of the homology (the alternating sum of traces; the alternating is necessary to work with long exact sequences). So, if we had a different operator on the stalks that was natural in the sense that it commuted with all long exact sequence differentials, then we would be cooking with gas.
That’s exactly what the Frobenius allows us to get. First of all, we need to consider objects with a bit more structure than just a sheaf. Rather, we need a sheaf that is equivariant for the Frobenius action, that is, with an isomorphism . For any sheaf which is somehow geometrically natural (for example, the constant sheaf for any abelian group), there’s no trouble doing this, though of course, one can always modify this action, say by scalar multiplication.
Now, this Frobenius structure may not seem so exciting, but it is rather important, because it commutes with everything you can possibly imagine. For example, if we have a short exact sequence of complexes of sheaves , then we have a long exact sequence relating the cohomology of these complexes
and all these maps commute with the Frobenius. In particular, if one looks at a point which is fixed by , that is, a point in , then there is an induced action of the Frobenius on the stalk at this point.
The function-sheaf correspondence
So at each point in , we get a map from the Grothendieck group of our category of sheaves to the base field k, given by the supertrace of on the stalks. Put another way, for each sheaf in our category, we get a series of functions , which only depend on the class of in the Grothendieck group. Think of this as a map of the Grothendieck group to the direct sum of functions on for all n.
I learned this from the book of Kiehl and Weissauer, but it’s much older than that. I’ll leave attribution to the comments.
Theorem. This map is injective.
So, this tells us a lot about the Grothendieck group of our category. But this theorem is most interesting because it’s functorial, in a certain sense. Let denote compactly supported pushforward of sheaves and the naive sum-over-fibers pushforward of functions on finite sets. Similarly, let denote the obvious pullback of sheaves, and the obvious pullback of functions on open sets.
Theorem. We have equalities , and .
So, lots of geometric operations on sheaves are easy to understand in this picture. Better yet, I’ve talked about the non-equivariant version, but I believe this will work just as well for morphisms between stacks, in which case the points will be a groupoid, not just a set, and the theorem above will say that pull and push on a correspondence is sent to the action of the corresponding span of groupoids on functions (this part, I’m less sure about, though I would be shocked if it wasn’t at least morally right).