Comparison theorems between algebraic and analytic De Rham cohomology (With emphasis on the p-adic case)

Abstract

We present a panorama of comparison theorems between algebraic and analytic De Rham cohomology with algebraic connections as coefficients. These theorems have played an important role in the development of D-module theory, in particular in the study of their ramification properties (irregularity…). In part I, we concentrate on the case of regular coefficients and sketch the new proof of these theorems given by F. Baldassarri and the author, which is of elementary nature and unifies the complex and p-adic theories. In the p-adic case, however, the comparison theorem was expected to extend to irregular coefficients, and this has recently been proved in [AB]. The proof of this extension follows the same pattern as in the regular case, but involves in addition a detailed study of irregularity in several variables. In part II, we give an overview of this proof which can serve as a guide to the book [AB].