Étale Cohomology

This is a post from here.

I have become very interested in algebraic geometry (only the terminology makes it kind of confusing), and I revised what little I knew (or used to know when I wrote the original post) about étale cohomology. Here I’ll try to give a brief introduction and some motivation as to why étale cohomology arose and how it’s been applied. I’ll be closely following Kiran Kedlaya’s notes on OCW MIT for this post.

Let f:Y\to X be a morphism of schemes. There is a canonical map \phi:\mathrm{Hom}_X(X^\prime,Y)\to\mathrm{Hom}_X(Z,Y) defined by any morphism g:X^\prime\to X and a closed subscheme Z defined by a nilpotent ideal of \mathcal{O}(X^\prime). If this map is always bijective then f is called formally étale. If f is also locally of finite presentation then we simply say f is étale.

It turns out that étale morphisms in algebraic geometry are like the notion of a covering space in topology! So we can define a cohomology theory, called étale cohomology. But there’s more motivation than this, namely the Weil conjectures.

Let X be a variety over a finite field \mathbf{F}_q. Define the zeta function as follows: \zeta_X(T)=\prod_x\left(1-T^{\mathrm{deg}(x\to \mathbf{F}_q)}\right)^{-1}.

The product is taken over the closed points of X. This zeta function isn’t too hard to compute, and it’s a nice exercise to write down the zeta function for X=\mathbf{P}^1 or X and elliptic curve (the latter is harder). Weil predicted that this zeta function can always be interpreted as the power series expansion of a rational function of T.

There’s something more significant than that, though. Look at the coefficients of the zeta function when X is an elliptic curve, for example. Then the degrees of the factors are 1,2,1. Where have we seen these before? These are the Betti numbers of a genus 1 Riemann surface! So Weil conjectured that if X is obtained from a smooth proper scheme over some arithmetic ring, reducing modulo a prime, then the degrees of factors of \zeta_X(T) correspond to the Betti numbers of (X\times \mathbf{C})^{an}.

Weil conjectured a cohomology theory \mathrm{H}^i_{Weil} for varieties over \mathbf{F}_q taking values in some finite dimensional verctor space over a field K of characteristic 0, such that the number of \mathbf{F}_q-rational points (fixed points of the q-power Frobenius map) can be computed by the following formula: \sum^{2\mathrm{dim}(X)}_{i=0}(-1)^i\mathrm{Trace}(\mathbf{F}_{q^n},H^i_{Weil}(X)), which implies the rationality of the zeta function.

It might seem like coherent sheaf cohomology might be useful. But apparently it’s not, for two reasons: it lives in characteristic p but not characteristic 0 (so we can prove rationality mod p), and dimensions don’t match Betti numbers. It’s now that Grothendieck came in; he decided to find an analogue of topological cohomology with étale maps, corresponding to local homeomorphisms (we now see how étale maps correspond to covering spaces). For example, from GAGA, every finite covering space map for a smooth proper variety X over \mathbf{C} comes from a unique finite étale cover of X. So the profinite completion of the topological fundamental group can be recovered as the inverse limit of the Galois groups of these étale covers (this example is taken straight from the lecture notes referred to above).

Now, Grothendieck was forced to modify the notion of a topology, leading to the notion of a Grothendieck topology. Let X be a topological space. Then a presheaf on X is simply a map \mathrm{Op}(X)^op\to\mathcal{C}. So one can state all the sheaf axioms in terms of that category!

Let \mathcal{C} be a category admitting finite products. A Grothendieck topology consists of coverings \{U_i\to X\} for each X\in\mathcal{C} satisfying the following conditions:

  1. X\xrightarrow{\sim} Y is a covering of X
  2. If for Y\to X, \{U_i\to X\} is a cover, then \{U_i\times_X Y\to Y\} is a cover
  3. If \{U_i\to X\} is a cover and for each i, \{V_{ij}\to U_i\} is a cover, then \{V_{ij}\to X\} is also a cover.

This is actually a Grothendieck pretopology, as Kedlaya notes, and one should take the coverings “generated” by this pretopology/basis. There are many other Grothendieck topologies, like the fppf, fpqc, smooth, flat, Nisnevich (which I’m trying to study with K-theory), and syntomic topologies.

Now I’m going to define the big and small étale sites, and I may rant a bit here. The big étale site \mathrm{Big}X of a scheme X is a subcategory of \mathrm{Sch}_{/X} spanned by the schemes over X with coverage given by étale covers. The small étale site is the subcategory X_{etale} of the big étale site whose objects are schemes whose structure map to X is étale. A covering in X_{etale} is a covering \{U_i\to U\} in the big étale site, such that U\in X_{etale}.

Let \mathcal{F}:\mathrm{Big}X^{op}\to\mathrm{Ab} be a presheaf on \mathrm{Big}X. A sheaf on this site is a presheaf such that for any cover \{U_i\to X\}, there is an exact sequence 0\to \mathcal{F}(X)\to \prod_i\mathcal{F}(U_i)\to\prod_{i,j}\mathcal{F}(U_i\times_XU_j).

There is also the notion of sheafification, and it becomes very interesting here because there’s no notion of a point. But, what is a point but a decreasing family of open sets? Points can be defined in that way, and that brings us to the concept of a topos. I won’t go into that here, but at least now there’s some more motivation for studying topoi.

We should really get around to defining étale cohomology. Let X be a scheme and X_{etale} its small étale site. Then let \mathcal{F} be an \mathrm{Ab}-valued sheaf on X_{etale}. The global sections functor, \Gamma(X,-) is left exact and the right derived functors of the global sections functor gives us étale cohomology.

Now, what about the Weil conjectures? Let q\neq p be a prime. Then define \mathrm{H}^i(X) (the Weil cohomology) as \mathrm{H}^i_{etale}(X\times_{\mathbf{F}_q}\mathbf{F}_q,\underline{\mathbf{Z}_q})\otimes_{\mathbf{Z}_q}\mathbf{Q}_q. These are finite dimensional \mathbf{Q}_q vector spaces, and they have a Lefschetz trace formula, implying rationality of the zeta function. I won’t go into much detail here because I think I’ve conveyed the general idea across.




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