It is true that p-adic cohomology theories for algebraic varieties defined over fields of characteristic p are more difficult, but often more useful, than ℓ-adic (ℓ different from p) étale cohomology. Rigid cohomology was defined by Berthelot as a common generalisation of crystalline cohomology for smooth proper varieties and the Monsky-Washnitzer cohomology for affine varieties. It is defined in terms of overconvergent de Rham cohomology of suitable characteristic zero lifts; this was exploited by Kedlaya and his followers who developed efficient algorithms that are of interest to cryptographers for computing the number of points on (certain) algebraic varieties defined over finite fields. This book, which is based for the most part on Berthelot’s unpublished preprints, is the first monograph on the subject. After a user-friendly introduction, the author explains the geometric background of overconvergence (tubes and strict neighbourhoods) in chapters 2 and 3. The following two chapters treat the analytic and sheaf-theoretical aspects of overconvergence. Chapter 6 studies overconvergent de Rham cohomology, which is reinterpreted in chapter 7 in terms of overconvergent isocrystals (i.e. analogues of lisse sheaves in ℓ-adic cohomology). Chapter 8 treats rigid cohomology of algebraic varieties with coefficients in an overconvergent (F-)isocrystal. The final chapter gives an informal overview of several more advanced aspects of p-adic cohomology. The book is well-written, with a mixture of concrete examples and abstract theory. It is accessible to readers familiar with basic concepts of abstract algebraic geometry and p-adic analytic (rigid) geometry.

Reviewer:

jnek