${\scr D}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini

Let p be a prime number, S a Z(p)-scheme, X -^ S a smooth morphism of schemes. Using a weaker form of th classical notion of divided powers on an idal, we construct on X an inductive System of sheaves of differential operators P^ , which are locally generated by divided powers of drivations up to order p^, and therefore hve better finiteness than th usual sheaf of differential operators. On a smooth p-adic formai scheme X, we aiso introduce th p-adic completion T>^ of V^ , and th sheaf T>\ = lim^n T>^ . We prove that th sheaves "D^ and T>^ g = T) ^ 0 Q are cohrent, and that theorems A and B hold for cohrent modules on thse rings. Finally, we construct analogous sheaves of differential operators with overconvergent singularities along a divisor, and show that overconvergent isocrystals can be viewed as cohrent modules over thse rings.