Arithmetic differential operators with overconvergent coefficients

Appears in collection : Tropical Geometry, Berkovich Spaces, Arithmetic D-Modules and p-adic Local Systems

The ring of differential operators with overconvergent coefficients plays a central role in Ber- thelot's theory of arithmetic D-modules over a formal scheme, as well as in Caro's theory of overholonomic modules. It is constructed over the complement of a divisor of some smooth formal scheme. We would like to explain how to do such a construction in the case where we have a non necessary smooth divisorial compactification of some smooth formal scheme.