Rigid Cohomology
Bernard le StumIn 2004, I was asked by Professor King Fai Lai to come to Peking University in
order to give a course on rigid cohomology. We agreed on the last two weeks of
January 2005. I want to thank here Professor Zhao Chunlai for the organization
of my visit as well as Professor Zhou Jian and his wife for showing us the city.
My family and I will always remember it.
While preparing this course, I realized that there was no introductory book
on rigid cohomology. Actually, there was no available material in English and
only an old document in French, Cohomologie rigide et cohomologie rigide
à support propre, by Pierre Berthelot. A revised version of the first part of
this document appeared as an official preprint in 1996 but the second part is
not fully written yet and, therefore, not really available to the mathematical
community. Fortunately, Berthelot was kind enough to answer my questions
on this second part and point out some articles where I could find some more
information.
Rigid cohomology was introduced by Berthelot as a p-adic analogue of
l-adic cohomology for lisse sheaves, generalizing Monsky–Washnitzer theory
as well as crystalline cohomology (up to torsion). Recently, it appeared that
this theory may be used in order to derive new algorithms for cryptography.
The first result in this direction is due to Kiran Kedlaya who has also done
incredible work on the theoretical aspect of the theory.
I knew that it was impossible to cover the full story in twenty one-hour
lectures. I decided to first introduce the theory from the cryptography point of
view(Introduction),thendescribethebasicsofthetheorywithcompleteproofs
(heart of the the course), and conclude with an overview of the development
of the theory in the last 20 years (Conclusion). In particular, the main part of
this book is quite close to Berthelot’s original document. I hope that this will
be useful to the students who want to learn rigid cohomology and, eventually,
improve on our results.