I’ve been attending a seminar/class run by Nick Proudfoot preparing for his workshop this summer on canonical bases. In conversations with Nick and graduate students, and there’s been some confusion about the relationship between Hall algebras and Grothendieck groups. Obviously, if you read the definitions you’ll see they are not the same, but the idea seems to be floating around that there is something going on with them. At some point, I decided writing a blog post on the subject would be a good idea. What are Hall algebras?
The Hall algebra of a category is the Grothendieck group of constructible sheaves/perverse sheaves on the moduli stack of objects in the category. The Hall algebra is an algebra because the constructible derived category of the moduli stack of objects in abelian category is monoidal in a canonical way.
To my mind, this is what makes Hall algebras worth studying, yet it’s oddly ignored in the literature on them (as far as I know; people should feel free to correct me). For example, it’s never mentioned in Schiffmann’s Lectures on Hall Algebras, the closest thing the subject has to a standard reference.
A bit more detail
So, let me unpack all of this. What people will usually tell you is that the Hall algebra is the functions on the set of objects of an abelian category. Important examples include the category of representations of a quiver and the category of coherent sheaves on a projective curve. So, when I say it actually has to do with the Grothendieck group of constructible sheaves on the moduli stack of objects, I have to explain two things:
- Why are the set of objects in an abelian category actually a stack?
- What do functions on a space have to do with the Grothendieck group of sheaves on said space?
Luckily, I’ve already addressed 2. in another blog post. Roughly, given a complex of sheaves with constructible cohomology, you can take the Euler characteristic of the stalk (or much better, the super-trace of Frobenius on the stalk) and get a function, which obviously only depends on your class in the Grothendieck group.
So the question is why the set of objects is secretly a stack; as usual, I’m sure I will get some technicalities wrong if I try to work in full generality, so let me restrict to the question of why a finitely generated algebra over a field has a moduli stack of finite dimensional representations. In this case, the space of -actions on has a very natural structure as an affine variety; pick generators , and create n-by-n matrices full of formal variables for each of them. Now, take the list of relations the satisfy; maybe . Look at the corresponding product of the ‘s (in our example ) and set the entries of that matrix to 0. That’s some polynomial equations in the entries of the matrix, and they define an affine variety whose points over any commutative -algebra are n-dimensional representations (with fixed basis!) of over that base.
But, I’ve overcounted! Lots of these are isomorphic representations with different choices of bases. Luckily all isomorphisms come from changing basis on , that is, acting by on by conjugating matrices. So really, I want ; but we have to think about what this means. It’s not a variety, sadly that’s not how quotients of varieties by groups work.
Instead, it’s a stack. This is really just a recognition of the fact that objects in categories aren’t just inert things; they have automorphisms (which in this quotient show up as the stabilizers of points). So this shows, modulo your willingness to accept the function/sheaf correspondence that the Hall algebra is the same thing as the Grothendieck group of sheaves on the moduli space of objects. Except, I haven’t said anything about the multiplication.
The monoidal structure
Where does the multiplication on the Hall algebra come from? Well, from a correspondence which looks like this:
The middle term is the stack of short exact sequences in our category with terms of dimension , and ; the maps are “take the outside terms” and “take the middle term.”
The Hall algebra multiplication is “pull back functions from the left, and pushforward to the right.” Of course, some care is necessary to make sure we interpret “pushforward” correctly. This is why people often require “finitary categories” for Hall algebras. Then it’s easy to say what “pushforward” should be; it’s just summing over the (finite) fibers of that map accounting for the that that points in come with automorphisms. There are also some funny powers of q (see Schiffman) which come from fixing things so that pushforward preserves Verdier duality (they arise directly from the virtual dimension of the components of the stack ).
EDIT: Joel points out that this notation is probably needlessly obscure. The point is that I’m looking at the stack of short exact sequences which has three projection maps to the stack of objects given by taking the sub-, total, or quotient objects. Multiplication in the Hall algebra is essentially (except with pullback adjusted by some funny constants from the Euler form). All I’m doing above is thinking of the stack of objects in more concrete terms. END EDIT
One of the points I at least tried to emphasize in my earlier posts on this stuff is that the Grothendieck trace formula really says that
The action of correspondences on categories of sheaves by push-pull categorifies their action on functions by push-pull
OK, so that’s not literally what Grothendieck said, but maybe it’s what he should have. What structure does the correspondence
induce on the level of categories of sheaves? Well, it lets one take sheaves on and , take outer tensor, pull and push, and get a sheaf (well, really complex of sheaves) on $R_A^d/GL_d$. It’s a monoidal structure on the derived category of sheaves on the sheav on the moduli space, assuming you think of sheaves on each component of the moduli space living together as a happy family. This structure was (for example) extensively studied by Lusztig, sadly, before the language of monoidal categories had gotten popular (well, actually is it popular yet? Anyways, it’s more popular now than in the 80’s). Anyways, I’ve taken the liberty of calling this the Hall monoidal structure for reasons which I think should be obvious.
So the upshot of all this discussion is:
Theoremesque: The Hall monoidal structure on sheaves on the moduli space of objects in a monoidal category categorifies the Hall algebra of the category. Similarly, all obvious subalgebras of the Hall algebra correspond to subcategories of the monoidal category.
I call this “theoremesque” because for a real theorem, I need to be a lot more careful about hypotheses. This isn’t so much a theorem as a principle. But it’s a good principle; it leads one to construct canonical bases and categorifications of quantum groups, but that’s a story for another post.