Problems with de Rham and l-adic

Let’s begin by reviewing the fact that when we work over ${\mathbb{C}}$ we have a nice comparison theorem. Last time it was briefly mentioned that if ${X}$ is smooth over ${\mathbb{C}}$, then we can consider the analytic topology on the ${\mathbb{C}}$-valued points which we’ll call ${X^{an}}$. Using GAGA and the degeneration of the Hodge-de Rham spectral sequence from last time we immediately get that ${\mathbf{H}^*(X, \Omega_{X/\mathbb{C}}^\cdot)\stackrel{\sim}{\rightarrow} \mathbf{H}^*(X^{an}, \Omega_{X^{an}}^\cdot)\simeq H^*(X, \mathbb{C})}$.

Thus we have a comparison theorem that allows us to use purely algebraic methods to recover the topological information of the singular cohomology. But what happens if we work over a field that is not ${\mathbb{C}}$? Well let’s assume that ${X}$ is smooth and of finite type over ${k}$. If ${k}$ has characteristic ${0}$, then the standard idea of the Lefschetz principle gives us that everything works out again and we can recover the singular cohomology.

Obviously lots can go wrong in positive characteristic. Say ${\mathrm{char}(k)=p}$, then we’ll throw out the case of affine things because for instance if ${X}$ is an affine curve the de Rham cohomology is not even finitely generated. But if we assume ${X/k}$ is proper and have finite generatedness, we still have problems with de Rham. It doesn’t recover the correct topological information. For instance, we’d like a cohomology that gives us “correct” Betti numbers.

If we define the Hodge Betti numbers to be ${\displaystyle b^i_H=\sum_{p+q=i}\dim_k H^q(X, \Omega^p)}$ and the de Rham Betti numbers ${b^i_{dR}=\dim_k H^i_{dR}(X/k)}$ we can just look at the HdR SS and see that ${b^i_H\leq b^i_{dR}}$ and we have equality if and only if the HdR SS degenerates at ${E_1}$. Degeneration of this spectral sequence is related to having some sort of analogue of ${X}$ in characteristic ${0}$, also known as a “lift” of ${X}$. Even if this spectral sequence degenerates, it merely relates these two Betti numbers, but doesn’t fully relate back to the topological Betti numbers.

Also, despite the fact that there is still a Lefschetz fixed point formula for de Rham cohomology in positive characteristic, it only gives the right answer mod ${p}$. We’ve seen that de Rham cohomology is a nice attempt at coming up with an algebraic way of defining singular cohomology (and is quite useful for lots and lots of things), but it seems very tied to smoothness and characteristic ${0}$ issues.

Anyway, it was good to point that out, but many of you might be frustrated because we already knew that de Rham was going to be a failure in characteristic ${p}$. If you’ve been introduced to ${\ell}$-adic cohomology, then you probably already realized that this was basically invented to solve some of these problems. Now we’ll impose the extra condition that ${k}$ be algebraically closed. Recall that we can just pick a prime ${\ell\neq p}$ and define ${H^i_\ell (X)=H^i(X_{et}, \mathbb{Z}_\ell)=\lim H^i(X_{et}, \mathbb{Z}/\ell^n)}$. This is a ${\mathbb{Z}_\ell}$-module.

Using the comparison theorem we see that if ${k=\mathbb{C}}$, then ${H^i_\ell(X)\simeq H^i(X^{an}, \mathbb{Z})\otimes \mathbb{Z}_\ell}$. This is quite nice. ${H^i_\ell(X)}$ tells us exactly the topological information we sought, since we recover the rank of ${H^i(X, \mathbb{Z})}$ and the ${\ell}$-primary torsion. Thus if we run through all primes ${\ell}$ we can completely recover ${H^i(X, \mathbb{Z})}$ up to isomorphism. Also, we see that the rank of ${H^i_\ell(X)}$ is finite and independent of ${\ell}$.

Here is the first of many problems. Although for a smooth projective scheme in positive characteristic we do get that the rank of ${H^i_\ell(X)}$ is finite, it is unknown if this is a number independent of ${\ell}$. The second much more severe problem is that we only have the reasonable properties listed in the previous paragraph for ${\ell\neq p}$. So even though we can get a bunch of topological information about ${H^i(X, \mathbb{Z})}$, we are unfortunately left clueless about the ${p}$-torsion. We also lose a bunch of classical theorems (even when ${\ell\neq p}$) such as the Lefschetz hyperplane theorem. Or if we consider the induced map from ${X\rightarrow X}$ on ${H^i_\ell(X)\rightarrow H^i_\ell(X)}$ there may be powers of ${p}$ in the denominator of the characteristic polynomial that we won’t ever be able to get information about.

Clearly, ${\ell}$-adic cohomology isn’t going to quite do the trick. All I seem to be doing is complaining about how these cohomology theories are failing what I want, but I haven’t ever told you exactly what it is that I do want from a cohomology theory. So next time we’ll start working on stating those properties and seeing if anything we’ve come across satisfies them and if not how we’re going to construct something that does.

4 thoughts on “Problems with de Rham and l-adic”

2. Is it really unknown if the rank of the $l$-adic cohomology of a smooth proper variety over a finite field is independent of $l$? By the Weil conjectures, we have that the Frobenius action on $H^i$ has eigenvalues which are algebraic integers of absolute value $q^{i/2}$. By the trace formula, the global zeta function of the variety (which is totally independent of $l$), is equal to the product of the determinants of $1 - tF$ on these cohomology groups (some in the numerator, some in the denominator). So it seems that if you look at the distribution of the roots of the zeta function (namely, how many there are on a given circle), you can recover the dimension of the $l$-adic cohomology, no?
Also, there is a Lefschetz hyperplane theorem in $l$-adic cohomology (proved in the same way Griffiths-Harris prove the one for complex cohomology: you use the fact that an *affine* variety of dimension $n$ has no cohomology in degrees $>n$ — in the complex world, this is because the variety has the homotopy type of a $n$-dimensional CW complex by Morse theory — and then using the Gysin sequence of the smooth pair $(X, X \cap H)$ (where $X$ is your smooth proj. variety, $H$ a hyperplane so that the section is smooth).
There’s even a hard Lefschetz in $l$-adic cohomology, but it was only proved in Weil II (and generalized by BBD)! Statement: if $X$ a smooth projective variety of dim. $n$, and $\eta \in H^2$ the first Chern class of a very(?) ample line bundle, then cupping with $\eta^{2i}$ gives an isomorphism between $H^{n-i}$ and $H^{n+i}$ (with some Tate twist).
As for the hard Lefschetz theorem for $\ell$-adic cohomology, I did learn about that since I posted this, so yes that does actually work. As for the independence statement, I’m pretty sure that when I posted this I meant to have these failings as things that we didn’t yet know on the blog and historically as motivation for crystalline cohomology. Rereading this post, I realize it doesn’t really come across that way.
This reminds me that I should probably understand why $\ell$-adic is independent of $\ell$ considering I actually use this fact sometimes.