Finitude de la dimension homologique d'algèbres d'opérateurs différentiels faiblement complètes et à coefficients surconvergents

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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.