Some Elements on Berthelot’s Arithmetic $${\mathscr {D}}$$ -Modules

Abstract

AbstractThis text is an introduction to Berthelot’s theory of arithmetic \(\mathscr D\)-modules. We first review Berthelot’s ring of differential operators of finite level on affine smooth p-adic formal schemes over a complete discrete valuation ring of mixed characteristic (0, p) with perfect residue field. Berthelot’s ring is a kind of weak p-adic completion of the usual ring of differential operators as defined by Grothendieck. We finish with the description and some finiteness properties of the constant coefficient which is constructed by adding overconvergent singularities. This lecture is suitable for graduate students and requires only basic knowledge of ring theory and algebraic geometry.