Un théorème de comparaison entre les faisceaux d’opérateurs différentiels de Berthelot et de Mebkhout-Narváez-Macarro

In this paper, we compare in one particular case, the arithmetic $D$-modules introduced by Berthelot and the arithmetic $D$-modules introduced by Mebkhout and Narváez-Macarro. We prove that there exists an equivalence of categories of coherent $D$-modules when you consider the arithmetic $D$-modules introduced by Berthelot on a projective smooth formal scheme $\mathcal {X}$, over some discrete valuation ring $R$ of mixed characteristics $(0,p)$, that is endowed with an ample divisor, along which the coefficients of the differential operators are overconvergent. On the side of Mebkhout-Narváez-Macarro, you have to look at differential operators over a smooth, affine, weakly formal scheme over $R$, whose $p$-adic completion is the complementary of $Z$ into $\mathcal {X}$. The equivalence of categories is given in one direction by taking global sections of the $D$-modules.