Etale Cohomology: Motivation

SEE EDIT AT BOTTOM OF POST.

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.

EDIT: 7 years later…

I have learned a lot since I wrote the above post 7 years ago. I’d like to write something which differs from the above on some points, and is the same on others, but hopefully will still be useful.

Why is the Zariski topology bad for…topology?

The Zariski topology is at its core a bookkeeping tool for algebra–a little bit like the ‘topology’ which shows up in Furstenburg’s proof of the infinitude of primes. Don’t get me wrong it’s a wonderful, intuitively powerful, essentially indispensable bookeeping tool, but a bookeeping tool nonetheless. The Zariski topology is not good at doing things that we think of ‘classical topology’ as being good at. To convince the reader of this, we can consider the following (interrelated) facts.

Fact 1: Let $X$ be an irreducible topological space. Then, every locally constant sheaf of sets is constant, and every constant abelian sheaf has trivial cohomology.

Proof: Let $\mathscr{L}$ be a locally constant sheaf. Since any two open sets must intersect, it’s easy to see that $\mathscr{L}$ must in fact be locally isomorphic to $\underline{S}$ for some set $S$. Then, by the standard theory of twists (e.g. see this post) it suffices to show that the non-abelian cohomology set $H^1(X,\mathrm{Aut}(\underline{S}))$ is a singleton. But, note that $\underline{S}(U)=S$ for every open subset $U$ of $X$, since $U$ is irreducible, and thus connected. But, this then implies that $\mathrm{Aut}(\underline{S})(U)=\mathrm{Aut}(S)$ for all open subsets $U$ of $X$. It is then easy to see by explicitly thinking about cocycles that $H^1(X,\mathrm{Aut}(\underline{S}))$ is a singleton as desired.

To see the second claim, let $A$ be an abelian group. We aim to show that $H^i(X,\underline{A})=0$ for all $i\geqslant 0$. But, again since for all open subsets $U\subseteq V$ the map $\underline{A}(V)\to \underline{A}(U)$ is a bijection, we know that $\underline{A}$ is flasque and thus has trivial cohomology (e.g. apply this with $\mathcal{O}_X=\underline{\mathbf{Z}}$). $\blacksquare$

Fact 2: Let $X$ be an irreducible topological space, then every connected covering space of $X$ is an isomorphism.

Proof: Let $\pi:Y\to X$ be a connected covering space. Consider the sheaf $\mathcal{F}$ of sections of $\pi$. Note then that $\mathcal{F}$ is locally constant, and thus by Fact 1, constant. This implies that there is, in particular, a global section $s:X\to Y$ of $\pi$.  But, $s$ is easily shown to be an isomorphism with inverse $\pi$. $\blacksquare$

And perhaps, most damning:

Fact 3: Let $X$ be an irreducible sober topological space. Then, $X$ is contractible.

Proof: Since $X$ is irreducible and sober there exists a generic point $\eta$ in $X$ such that $\overline{\{\eta\}}=X$. Note then that

$H:X\times [0,1]\to X,\qquad H(x,t):=\begin{cases}\eta & \mbox{if}\quad t\ne 0\\ x & \mbox{if}\quad t=0\end{cases}$

is a deformation retract of $X$ onto $\{\eta\}$. $\blacksquare$

All of these properties were about irreducible topological spaces. Irreducibility is a useful notion in the context of an algebraic bookkeeping tool, it tells us roughly when an algebraic-y space is in ‘one piece’, but it is clearly a bad property to have interesting ‘algebraic topology’.  But every Noetherian scheme (which we are almost always happy to work with in algebraic geometry) is a finite union of irreducible sober subspaces (e.g. see this).

From all of this we deduce the following:

Moral: The open sets in the Zariski topology are too big to be useful for classical topological study.

A hint from complex geometry

So, to fix the problems with the Zariski topology one needs to add in ‘smaller open subsets’. Of course, the problem with this is the obvious one: algebraic open subsets are necessarily large! For example, if $X$ is a smooth variety over $\mathbb{C}$, the kind of open subsets we want are things like small open disks in the analytification (e.g. see Expose XII, SGA 1) $X^\mathrm{an}$–that would give us the ‘correct topology’. But, such small open subsets are never algebraic, again because Zariski open subsets have to be so large (e.g. their complements have measure zero). So, we cannot get arbitrarily small open subsets of $X^\mathrm{an}$ algebraically.

The etale topology allows us to fix this issue, but not in the most obvious way. A misconception that I once had is that somehow the etale topology allowed us to get algebriac morphisms which were actually ‘smaller’ (in some sense) than Zariski opens. But, as already mentioned, the largeness of algebraic-y objects is inescapable. Instead, the etale topology gets at the `correct topology’ of a scheme in a more roundabout way.

To help motivate this, we again consider the case of a variety $X$ over $\mathbb{C}$ which, for comfort’s sake, we will assume is smooth. In this case we know that the ‘correct topology’ we wish to access is the topology of $X^\mathrm{an}$. As we saw in Fact 1 above, one of the main issues with $X$ is that its sheaf theory is not ideal. So, we would like to somehow access the sheaf theory of $X^\mathrm{an}$ algebraically.

To state the key observation to explaining how this is possible (or at least partially so), let us fix $M$ to be a complex manifold. Define by $M_\mathrm{et}$ the category for which:

• the objects are lolcaly on the source biholomorphisms (also known as etale morphisms) $M'\to M$,
• a morphism $M'\to M''$ is a map of complex manifolds over $M$.

We would like to endow $M_\mathrm{et}$ with the structure of a ‘Grothendieck (pre)topology’ (I won’t define/motivate this definition here, but this is done nicely in Wikipedia). Namely, for an object $M'$ of $M_\mathrm{et}$ let us define its set of covers to be sets of the form $\{M'_i\to M'\}$ which are jointly surjective on the underlying topological space.

Let us note that one has a functor

$\mathbf{Sh}(M_\mathrm{et})\to \mathbf{Sh}(M),\qquad \mathcal{F}\mapsto \mathcal{F}|_M$

where $\mathcal{F}|_M$ means only consider the values of $\mathcal{F}$ on those $M'\to M$ which are open embeddings–this is clearly equivalent to giving a sheaf on the topological space $M$.

We then have the following (whose proof is left as an easy exercise to the reader).

Fact 4: Let $M$ be a complex manifold. Then, the functor $\mathbf{Sh}(M_\mathrm{et})\to\mathbf{Sh}(M)$ given by $\mathcal{F}\mapsto \mathcal{F}|_M$ is an equivalence.

So, we see that to get at the correct notion of sheaves on $X^\mathrm{an}$, we actually don’t need to define ‘smaller open subsets’ of $X$, but instead only try suss out when a map $Y\to X$ is a ‘local biholomorphism’ (i.e. etale). This has a fighting chance of being more useful, since there are many more algebraic ‘local biholomorphisms’  then there are actual Zariski open embeddings (e.g. the squaring map $\mathbf{G}_{m,\mathbf{C}}^\mathrm{an}\to\mathbf{G}_{m,\mathbf{C}}^\mathrm{an}$).

Moral: Instead of defining ‘open embeddings smaller than Zariski open embeddings’ we should instead try to define when a morphism $Y\to X$ is a ‘local biholomorphism’.

Etale morphisms and their desiderata

So to stop burying the lead: etale morphisms are precisely meant to answer the moral cry from the end of the last section–they are mean to be the algebraic analogue of ‘local biholomorphism’.

There are many ways to define when a morphism of schemes is meant to be etale. But, we just choose one here. Namely, let us say that a morphism of schemes $Y\to X$ is etale if it’s locally of finite presentation, flat (e.g. see here), and unramified (e.g. see here). These are the correct analogues of ‘local biholomorphism’ in algebraic geometry. I won’t spell out the intuition for this here (it can be extracted quite nicely from the previous two links), and instead list some of the properties of etale morphisms which show that such morphisms ‘act’ like local biholomorphisms.

• Open embeddings are etale.
• Etale morphisms are open maps (e.g. see this, or Theorem 12 of this).
• Etale morphisms are closed under composition (see this) and base change (see this).
• An etale and universally injective morphism is an open embedding (see this). As a slogan one might imagine “etale morphisms are like non-injective open embeddings”.
• If $Y\to X$ is a morphism of schemes and $U\to X$ is an etale surjection, then if $Y_U\to U$ is etale, then $Y\to X$ is etale (see this). As a slogan one might imagine “etale morphisms are local in the ‘etale topology’ on the target”.
• A morphism of schemes $f\colon Y\to X$ is etale if and only if for all $y$ in $Y$ there exist affine open neighborhoods $\mathrm{Spec}(B)$ of $y$, and $\mathrm{Spec}(A)$ of $f(y)$ such that $f(\mathrm{Spec}(B))\subseteq \mathrm{Spec}(A)$ and one has an isomorphism $B\cong A[x_1,\ldots,x_n]/(f_1,\ldots,f_n)$ where $\det(\frac{\partial f_i}{\partial f_j})$ is an invertible element of $A$. As a slogan one might imagine “etale morphisms are defined so to satisfy the setup of the inverse function theorem”.
• If $f\colon Y\to X$ is a morphism of smooth finite type $\mathbf{C}$-schemes, then $f$ is etale if and only if $f^\mathrm{an}$ is a biholomorphism (see Expose XII, Proposition 3.1 in SGA 1).

All of these properties strongly suggest that etale morphisms are as written on the tin: algebraic analogues of local biholomorphisms (i.e. etale morphisms) in complex geometry.

So, does it work? Does the story end nicely with the theory of etale morphisms–is there a good theory of sheaves? The answer is yes, but this is a very long story. For a rapid but lucid discussion of this I would suggest reading Deligne’s SGA 4.5. But, let me at least try to tie this discussion back into the desire to get the category of sheaves on $X$. Namely, suppose that $X$ is a finite type smooth $\mathbf{C}$-scheme. Then, there is a morphism of categories of sheaves $\mathbf{Sh}(X^\mathrm{an}_\mathrm{et})\to \mathbf{Sh}(X_\mathrm{et})$. Unfotunately, it can’t be an equivalence (e.g. the lefthand side contains (the sheaf represented by) open disks in $X$), but it does induce an equivalence $\mathbf{Loc}(X_\mathrm{et}^\mathrm{an},\Lambda)\to \mathbf{Loc}(X_\mathrm{et},\Lambda)$ between the category of $\Lambda$-local systems where $\Lambda$ is a finite ring, and this in turn gives rise to an isomorphism $H^i(X_\mathrm{et},\underline{\Lambda})\cong H^i(X_\mathrm{et}^\mathrm{an},\underline{\Lambda})$