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In this paper, we compare in one particular case, the arithmetic

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In this paper, we compare in one particular case, the arithmetic D-modules in- troduced by Berthelot and the arithmetic D-modules introduced by Mebkhout and Narvaez-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 schemeX, over some discrete valuation ringRof mixed charac- teristics (0, p), that is endowed with an ample divisor, along which the coefficients of the differential operators are overconvergent. On the side of Mebkhout-Narvaez-Macarro, you have to look at differential operators over a smooth, affine, weakly formal scheme overR, whosep-adic completion is the complementary ofZ intoX. The equivalence of categories is given in one direction by taking global sections of the D-modules.

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