Finitude de la dimension homologique dʹalgèbres dʹopérateurs différentiels faiblement complètes et à coefficients surconvergents

We prove that the sheaf of arithmetic differential operators with overconvergent coefficients, introduced by P. Berthelot, has finite cohomological dimension. A similar geometrical proof shows that the weak p-adic completion of the Weyl algebra has also finite cohomological dimension. Moreover, this algebra can be naturally endowed with a filtration which is compatible with the Frobenius.