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.
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In Strasbourg Part 1, I promised to deliver a few summaries of the minicourses given at the special week hosted at the Institut de Recherche Mathématique Avancée. In the next few posts, I will highlight a few things that occurred in Bernard le Stum‘s lectures on rigid cohomology. My posts are not meant to be a complete summary, and indeed I would like to be as succint as possible. Furthermore, I intend to add additional material explaining some of of the prerequisites. This first post covering rigid cohomology will try to explain the motivation for considering rigid cohomology in the first place.

Introduction

The lectures on rigid cohomology were given by Bernard le Stum who wrote the only existing textbook on rigid cohomology developed primarily by Pierre Berthelot, so the reader should definitely consult le Stum’s book for more details, as well as consult the slides when they appear, which contain many wonderful examples that I am omitting.

But what is rigid cohomology, and why is it useful?
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