Etale Cohomology: Motivation

In this series of posts, I will be documenting some aspects of étale cohomology, as I, myself, learn it. This will include a mixture of intuition, technical background, and examples. I will start with material that many readers may already be familiar with, through a course in algebraic geometry–topics such as flatness, smoothness, étaleness, etc.

There is no guarantee, and in fact it isn’t likely, that there will be anything here not present at some other place on the internet. There is also a high probability that some of what I say may be incorrect, either technically or intuitively. That said, I hope that some of my scribblings will be of use to some future learner of this brilliant and beautiful subject.

I will be following several sources, but most seriously will be Lei Fu’s Etale Cohomology, Milne’s Etale Cohomology, and SGA 4.5.

Motivation

Like many places in math, there are many natural reasons why etale cohomology (or other suitable replacements for the Zariski topology) is a desirable thing to study. But, in our specific case, there is a fairly definitive source from which the notion, or at least the desire for the notion, of etale cohomology arose. These are the Weil conjectures.

At the very heart of the Weil conjectures is the want of something relatively mundane: to count the number of solutions to a set of polynomials over a finite field. Or, in other words, to count the \mathbb{F}_q-points of some (projective) V defined over, say, \mathbb{F}_p. While this is an impossible question to answer in general, there are certainly some properties one might wish to deduce about the set of numbers N_m:=\#V(\mathbb{F}_{q^m}). In particular, one might hope to get some handle on the generating function associated to the N_m, or some variant thereof. This was what Weil wanted, but it was unclear how one might proceed.

Weil had a fairly brilliant, if not strange, idea. Namely, Weil noted that the set V(\mathbb{F}_{q^m}) was precisely the set of fixed points of the Frobenius map \text{Frob}_{q^m} acting on V(\overline{\mathbb{F}_p}). He then argued that one may use this observation, along with some analogue of the Lefschetz fixed point theorem, to actually get a handle on the N_m. Just to recall, the Lefschetz-Hopf theorem says that if X is a compact, triangularizable space, and f:X\to X is a continuous map, then

\displaystyle \sum_{x\in\text{Fix}(f)}i_f(x)=\sum_i (-1)^i \text{tr}\left(f_i^\ast\right)

where i_f(x) is the index of the fixpoint x, and f^\ast_i is the induced map H^i(X,\mathbb{Q})\to H^i(X,\mathbb{Q}).

While this is a unique idea it is obviously, on the surface, doomed. In particular, if V is irreducible, then one can show that for each i>0 the equality H^i(V,\mathbb{Q})=0 holds. This holds because every constant sheaf on V will be flasque, and since (under suitable conditions which hold in our case) one has that H^i_\text{sing}(V,\mathbb{Q})=H^i(V,\underline{\mathbb{Q}}) where the left is singular cohomology, and the right is sheaf cohomology. In fact, one can show that every irreducible scheme is contractible.

While this last result is, if you haven’t seen it before shocking it isn’t all that unexpected. The Zariski topology is a topology of convenience. It allows us to phrase purely algebraic arguments couched in the guise of topology. But, when we get right down to it, the Zariski topology is not the right topology in which to phrase more intricate geometric arguments, like counting fixed points topologically–it’s entirely too coarse.

The idea is then the obvious one. If the Zariski topology isn’t the right notion of a topology on a variety, then let’s create a new one. Let’s find some way of talking about the “right” (or one of the “right”) notions of a topology on V which plays behaves more in tune with our intuition. Of course, this is much easier said then done. It’s not at all obvious what properties this topology should hold (besides wanting to make the trace formula work), let alone how to describe it.

There is one relatively obvious property we would want to hold. While the Zariski topology is inadequate in many ways, there is a class of varieties which already comes with a “correct” topology. In particular, to every variety X/\mathbb{C} we have the associated analytic variety X^{\text{an}}. This is the space, with the topology, that we usually picture schemes as having. We are often even tempted (or at least I am) to argue, rigorously or analogically, theorems about X by using X^{\text{an}}. Thus, one property one might hope our “right” topology on X has is that it resembles, in some way, the topology of X^{\text{an}}.

It took decades before the algebraic geometry community as a whole (key roles being played by obvious people such as Grothendieck and Deligne) were able to hammer out this “right” topology, and bring to fruition Weil’s dream of using topology to solve problems over finite fields. As might be suggested by the length of time it took to create this topology, and who the main players were, it’s not shocking that the final result is one of sheer brilliance, initially overwhelming abstractness, and eventual it-should-have-been-clear-from-the-start-ness (all the makings of a great theory!).

The key insight made by those founders of etale cohomology was the realization that the topology we were looking for, was never meant to be a topology at all–or at least not one in the conventional sense. In particular, there are far too few subsets of some varieties, say ones over finite fields, to ever hope to apply “usual topological” techniques. The realization was, in line with Grothendieck’s relative view of the world, that the actual open sets didn’t live inside of the spaces themselves, but as covers of the space.

In particular, there was the realization that the open embeddings U\hookrightarrow X into a scheme X weren’t carrying enough information, but that perhaps if we could recover the “missing” (invisible) information by relaxing precisely what types of maps into X we allow. If we allow not only isomorphisms onto an open subset of X, but “local isomorphisms” onto a subset of X maybe this set of maps will give us relevant “topological” information missing from just open embeddings.

While the resulting objects we get aren’t a topology on X (as we said, topologies in the conventional sense are doomed in this endeavor), they are generalizations thereof. And while they lack corporeal property of being literal subsets of X they still possess enough structure to define sheaves. Namely, we will define a “cover” of X in this new notion of a topology as follows: it’s just a set of these “local isomorphisms” \{V_\alpha\to X\} whose images jointly cover X. A \mathcal{C}-valued sheaf will then by a functor from this category of “local isomorphisms” over X (the morphisms being the obvious ones–morphisms relative to X) such that they have the unique gluing property with respect these “open covers”. What this will mean will later be clear, but roughly we can make sense of this since we can still make sense of the exact sequence defining sheaves.

Of course, all of this entirely just intuitive. It takes an immense amount of technical machinery to correctly define and buildup all of the above notions: “local isomorphisms”, “open covers” in this category of local isomorphisms, sheaves, cohomology thereof. Hopefully, in the upcoming posts, we will provide a rough sketch of how to develop all of these notions and eventually, put them all together to form a coherent theory.

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