Strasbourg 2012 Part 3: More Rigid Cohomology

In Strasbourg Part 2, I gave a bit of motivation for rigid cohomology, but I skirted defining anything substantial, except for the zeta function. Recall that we have an smooth algebraic variety $ X$ of pure dimension $ d$ defined over the finite field $ \mathbb{F}_q$, and initially we were interested in the rational points $ X(\mathbb{F}_q)$ of $ X$. This led us to define the zeta function

$ Z(X,t) = \mathrm{exp}( \sum_{r=1}^\infty N_r(X)t^r/r)$.

of $ X$. Furthermore, I asked the reader to have faith that there are rigid cohomology spaces $ H_\mathrm{rig}^i(X)$ of $ X$, that the Frobenius extends to an operator $ F$ on each of them, and that we have a product expansion

$ Z(X,t) = \prod_{i=0}^{2d}\mathrm{det}(1 – tq^dF^{-1}|_{H_\mathrm{rig}^i(X)})^{{-1}^{i+1}}$.

Obviously many details have been left out, but this will suffice for continuing. I should state the disclaimer that from now on things will be a bit more sketchy since I’m not familiar with the more specialized material, and eventually I will just give references. As I have mentioned previously, I am not an expert in these areas, and I ask the patience of the reader since these summaries may lack some of the polish of my usual posts.

Rigid Cohomology, Definition 1

This first definition might be the quickest way to get to after workable definition of rigid cohomology for our variety $ X$ over $ \mathbb{F}_q$. In this definition we suppose that there is a scheme $ \mathcal{X}$ over $ W(\mathbb{F}_q)$ such that $ X \cong \mathcal{X}\times_{W(\mathbb{F}_q)} \mathbb{F}_q$. We also have to suppose another somewhat more technical condition: we require the existence of a formal scheme $ \overline{\mathcal{X}}$ such that there is some relative normal cross divisor $ D$ with smooth components on $ \overline{\mathcal{X}}$ with $ \mathcal{X}$ being the complement of $ D$. In some sense, this conidition controls the geometry of the geometric and special fibers (more on this later!).

In this case, we can define the rigid cohomology of $ X$ to be

$ H_\mathrm{rig}^*(X) = H_\mathrm{dR}^*(\mathcal{X}\times_{W(\mathbb{F}_q)} \mathbb{Q}_q)$.

From this definition, one can already start computing rigid cohomology in simple cases, because now rigid cohomology is just the cohomology of a complex of modules over $ \mathbb{Q}_q$. The difficult part is computing the action of Frobenius. Here the action of Frobenius on functions will be the iterated Frobenius, which is raising functions to the $ q$th power. However, recall that in the way we’ve defined rigid cohomology, we are doing our computations with $ \mathcal{X}$, and the Frobenius action does not in general lift to $ \mathcal{X}$.

However, all is not lost, and we can replace $ \mathcal{X}$ with its $ p$-adic completion $ \widehat{\mathcal{X}}$, and this allows us to compute our lift of Frobenius (if there is enough interest I will explain this further along with an example). If we do this, we get a new problem: unless $ \mathcal{X}$ is proper, we cannot guarantee that the cohomology will be the same.

However, luckily there is an intermediate object between $ \mathcal{X}$ and $ \widehat{\mathcal{X}}$ called the weak completion $ \mathcal{X}^\dagger$, which allows us to lift Frobenius and keep the same cohomology. Even more luckily, we can still compute all this relatively easily, and this has been applied for instance by Kedlaya in the case of hyperelliptic curves.

Towards Cohomology

While computational satisfying, Definition 1 has some serious issues with regard to theoretical questions. First of all, we made all sorts of choices in defining the rigid cohomology of the variety $ X$ over $ \mathbb{F}_q$. In particular, although we “defined” rigid cohomology, a priori we don’t really know that Definition 1 consitutes a valid cohomology theory. If we could do this, then we would have cohomological machinery for theoretical purposes, and our Definition 1 would be handy for computations.

To do this, we start with a bit of general setup and definitions. We let $ K$ be a nontrivial complete ultrametric field of characteristic zero. Ultrametric just means that its valuation satisfies the strong triangle inequality: $ |a + b| \leq \max\{a,b\}$. We let $ k$ be the residue field of $ K$ and $ \mathcal{V}$ the corresponding valuation ring of $ K$. We also denote the maximal ideal of $ \mathcal{V}$ by $ \mathfrak{m}$.

Our goal will be to define rigid cohomology of $ X$ using a construction called the tube of $ X$. We first let $ \mathcal{AN}/K$ be the category of analytic varieties in the Berkovich setting. We fix some embedding $ X\hookrightarrow P$ from $ X$ into some formal scheme $ P$, and let $ P_K$ be its generic fiber in the category of analytic varieties. The specialization map $ sp:P_K\to P$ is then defined locally by $ x\mapsto \mathfrak{p} = \{ f : x(f) < 1\}$. The tube of $ X$ with respect to $ P$ is then defined to be

$ ]X[_P = \{ x\in P_K : sp(x) \in X\}.$

If $ X\hookrightarrow P$ is a closed embedding of a proper algebraic variety into a smooth formal scheme $ P$ then we can define

$ H_\mathrm{rig}^*(X) = H^*_{\mathrm{dR}}(]X[_P)$

In fact, this definition will reduce to the first case when $ X = P_k$. If $ X$ is not a proper variety, this is a still a cohomology theory, but won’t be the rigid cohomology in general, but rather convergent cohomology. Now, this may seem a bit more abstract, but it motivates the most general definition. Of course, this definition also is made with a choice of closed embedding $ X\hookrightarrow P$, but there is a direct proof that this is independent of $ P$ using an algebraic Poincare lemma and the so-called weak fibration lemma.

Going Further

From here on, understanding the more general definition of rigid cohomology requires the definition and a bit of discussion on the category of overconvergent isocrystals. Since I don’t feel fluent in this area yet, I will keep the remaider of this post to discuss various references where the interested reader can find more.

The first reference to consult is le Stum’s book Rigid Cohomology. However, le Stum’s book requires some background in rigid geometry, and one book that covers some of this material is J. Fresnel’s and M van der Put’s Rigid Analytic Geometry and Its Applications.

For those interested in reading a bit about how rigid cohomology can be used to prove the Weil conjectures, Kedlaya’s paper Fourier transforms and p-adic “Weil II” is a good start. For defining cohomology via algebraic de Rham cohomology in the case of crystalline cohomology, P. Berthelot’s and A. Ogus’ paper F-isocrystals and de Rham cohomology I might be a nice addition to the coffee table.

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