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The Grothendieck trace formula as categorification, I: the category and the comparison theorem February 12, 2009

Posted by Ben Webster in Algebraic Geometry.
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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 p, 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).

Local systems

Now, the kind of sheaves we’d like to work with are “constructible” ones. Let me explain these for a variety over \mathbb{C} first. The most important example of a constructible sheaf is the sheaf \underline{\mathbb{C}}_X 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:

  1. 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.
  2. We could instead take a locally closed subvariety Y\subset X and look at the sheaf that assigns to U\subset X the space of locally constant functions (or flat sections of a bundle) on U\cap Y.

Why are these sheaves interesting? Well, over \mathbb{C}, 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 \mathbb{C}, the constant functions on that set. We have just proved:

Proposition. The sheaf of locally constant functions \underline{\mathbb{C}}_X is quasi-isomorphic to the DeRham complex (cochain complex) as sheaves. In fact, the sheaf cohomology H^*(X;\underline{\mathbb{C}}_X)=\mathrm{Ext}(\underline{\mathbb{C}}_X,\underline{\mathbb{C}}_X) 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 \mathbb{F}_q, and I talk about the sheaf \underline{\mathbb{C}}_X on this variety, I am not just making stuff up; I have precise mathematics in mind, even if I haven’t explained it fully.

Constructible sheaves

So, I want a category of sheaves: for now let’s start with X, and cut it into pieces X_i (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 X_i 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 \{X_i\}-contructible sheaves on X, 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 A_\infty 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 X_i‘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 \mathbb{F}_q is a bit hard for geometrically/topologically minded people, but the amazing thing is, they really aren’t that different from varieties over \mathbb{C}, in a way that we’ll make precise shortly.

So assume that everything you did (including the definition of the X_i) 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 X_{\mathbb{C}} over \mathbb{C} and one X_{\bar{\mathbb{F}}_q} over \bar{\mathbb{F}}_q, given by reducing all of those polynomials mod p, along with pieces (X_i)_{\mathbb{C}} and (X_i)_{\bar{\mathbb{F}}_q}. (Note: I’m using the algebraic closure \bar{\mathbb{F}}_q now).

Interestingly, one can makes sense of the idea of a local system on a variety over \bar{\mathbb{F}}_q, using this fancy etale stuff. Though, it’s important to note, my sheaves are still of vector spaces over {\mathbb{C}}; 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 (X_i)_{\mathbb{Z}} (for example, you should just take the constant sheaf  \underline{\mathbb{C}}_{X_i}).   These have extensions by base change to (X_i)_{\mathbb{C}} and (X_i)_{\bar{\mathbb{F}}_q}

The derived categories of \{X_i\}-contructible sheaves that use only these local systems on X_{\mathbb{C}} in the complex topology and X_{\bar{\mathbb{F}}_q} 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 \bar {\mathbb{Q}}_\ell and \mathbb{C}. 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 \mathrm{Ext}(\underline{\mathbb{C}}_X,\underline{\mathbb{C}}_X), this is the same whether you compute it over \bar{\mathbb{F}}_q or \mathbb{C} (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.

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Comments

1. dt - February 12, 2009

I think the theorem is not quite right. E.g. local systems on the complex line and on the F_q line are very different. Things are better for complete things, but even there there are differences: in the complex topology the monodromy operators are allowed to have infinite order, and in the etale topology they are not. When you have a stratification you have to apologize somehow for the lack of wild ramification on the complex side. And so on?

To get a correct statement, I think you should fix some local systems on the strata of your complex thing, find the corresponding local systems on the strata of your characteristic p thing, and stipulate that you only care about sheaves you can build (or “construct”) out of these. Then there is an equivalence of categories. Or else it’s even worse than I think.

2. Matthew Emerton - February 12, 2009

Dear Ben,

One should probably mention Grothendieck’s name, and also
Mike Artin’s, in relation to the theorem at the end of your post.
As you know, the whole etale machinery was set up by Grothendieck,
and in particular, the theorem identifying singular and etale
cohomology for varieties over C is due to him.

The proper base-change theorem is also crucial for the theorem
you mention, I think, and is due to Mike Artin. (I think Deligne’s
local acyclicity theorem for vanishing cycles is also crucial, but
you already mentioned his name.)

Anyway, my point is more just to say that for those who are new
to the magic of etale cohomology (which is certainly one of the
more incroyable things in mathematics), this is one of Grothendieck’s
great legacies, and that in setting up the machinery, most of
the heavy lifting was done by him, Artin, and Deligne.

Cheers,

Matt

P.S. I assume that the C coefficients are just a secret way of saying
Q_l without causing offense to non-arithmeticians.

3. Emmanuel Kowalski - February 13, 2009

Is your F_q (in the main theorem) really meant to be the finite field, or should it be an algebraic closure?

4. Geordie - February 13, 2009

Dear dt,

in the complex topology the monodromy operators are allowed to
have infinite order, and in the etale topology they are not.

Is this true? For example I understood it is possible to obtain the local system on \mathbb{A}^1 \setminus \{ 0 \} with monodromy

11
01

as a local system, by taking the limit of the corresponding local systems with values in \mathbb{Z} / \ell^m \mathbb{Z} (which have finite order).

This is something that I have wanted to understand for a while.

5. Ben Webster - February 13, 2009

dt-

Fixing a set of local systems first (the easy thing to do is assume that they’re defined over the integers, thus avoiding the bad local systems in both the complex topology and over characteristic p) is exactly what you need to do, and if I had reread BBD before writing the post, I would have said it the first time. Thanks for pointing that out.

Matthew-

Yes. It’s also that in the stuff I’m going to talk about that subtlety isn’t terribly prominent. I’ll mostly just be talking about the case where all Frobenius eigenvalues are rational, which makes the difference between \mathbb{C} and \bar{\mathbb{Q}}_\ell less noticeable.

Emanuel-

Right. Another stupid thing I would have noticed if I had reread my source. Fixed.

6. Ben Webster - February 13, 2009

Geordie-

Your example isn’t bad enough, I think. Look at page 149 of BBD. It says

Pour le \mathbb{Q}_\ell faisceaux, \epsilon^* est seulement pleinement fidele. Sur une variete normale connexe X, munie d’un point base x\in X(\mathbb{C}), un faisceaux localement constant \cal F de \mathbb{Q}_\ell-espaces vectoriels de rang fini est dans l’image essentielle si et seulement si l’action de \pi_1(X(\mathbb{C})) sur {\cal F}_x un reseau.

I am not a French speaker, but my reading of that is

For \mathbb{Q}_\ell sheaves, \epsilon^* is only fully faithful. On a connected normal variety X, with a base point x\in X (\mathbb C) , a locally constant sheaf of \cal F of \mathbb{Q}_\ell -vector spaces of finite rank is in the essential image if and only if the action of \pi_1 (X (\mathbb {C})) on {\cal F}_x stabilizes a lattice.

So, the problem isn’t sheaves with unipotent monodromy, but ones with, say, transcendental monodromy.

7. Richard - February 13, 2009

Great post! I’m looking forward to the rest of the series.

8. Geordie - February 14, 2009

Ben –

yes, but the point that I was trying to make was the infinite order does appear to be possible (the local system I wrote down does stabilize a lattice).

Thanks for pointing out pg. 149. The bit before it is interesting too: this _is_ an equivalence with finite coefficients (e.g. \mathbb{Z} / \ell^m \mathbb{Z}).

9. Ten constructions of the cohomology of varieties « Embûches tissues - February 14, 2009

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10. Jon - February 14, 2009

Ben,

Thanks for this. Can you suggest a place to learn more about the isomorphisms $\mathbb{C} \simeq \overline{\mathbb{Q}_\ell}$?

11. dt - February 14, 2009

This pro-stuff is confusing, but Geordie must be right. Probably the formal definition of “local system of Q_\ell-modules” will be equivalent to giving a continuous homomorphism from the etale fundamental group to GL_n(Q_\ell). The etale fundamental group is compact, though, so its image must also be. It means that we are allowed unipotent monodromy, and other \ell-adic integer units as eigenvalues, but not monodromy with eigenvalue 1/\ell.

I guess this is what Ben’s translation is saying, also. The maximal compact subgroups of GL_n(Q_\ell) are those that preserve a lattice. (But maybe the word “transcendental” is the wrong one in this context? Some elements of Z_\ell^\times must correspond to transcendental numbers under the embedding in C.)

12. Ben Webster - February 15, 2009

But maybe the word “transcendental” is the wrong one in this context?

Right, that was one of those points where working over \mathbb{C} infected my thinking. There is a similarly flavored but somewhat different problem with the Riemann-Hilbert correspondence where D-modules defined over the rationals might have analytic solutions which are not algebraic.

The compactness argument is probably the slickest way to think about it, though not necessarily congenial for those not used thinking \ell-adically.

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15. Masoud - February 28, 2011

Hello Ben,

The theorem you state about injectivity of a certain map is proved in Laumon’s paper (Fourier Transform and Weil Conjectures). The proof uses Cebatorav density and the theory of perverse sheaves. I’m not sure if there is a more elementary proof.

I’m a little bit puzzled about your last paragraph about function-sheaves correspondence on stacks. Let us consider a simple example: Let G be an algebraic group over a field. Consider the stack BG=pt/G, and an etale sheaf F on this guy. The trace of frobenius function attached to F should be a function on BG(F_q)= pt/G(F_q); in other words, it should be a function on the groupoid of G(F_q) torsors (over a point). I’m not sure how to make sense of such an object. I’d be interested to know if anybody has an idea.

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