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A comparison theorem between the Berthelot and the Mebkhout-Narváez-Macarro fibre bundles of differential operators. (Un théorème de comparaison entre les faisceaux d’opérateurs différentiels de Berthelot et de Mebkhout-Narváez-Macarro.) (French) Zbl 1053.14015
The author compares the arithmetic \(D\)-modules introduced by Mebkhout and Narváez Macarro with the \(D\)-modules introduced by Berthelot. She proves that in the special case, when considering weakly differential operators over the weak completion of the complement of an ample Cartier divisor on the one hand and the ring of arithmetic differential operators over \(p\)-adic completion of the weakly formal smooth scheme with overconvergent coefficients along the divisor on the other hand, there exists a natural equivalence of the categories of coherent modules. The proof is difficult but the author gives very thorough and precise presentation.

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14G20 Local ground fields in algebraic geometry
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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[1] A. Grothendieck, en collaboration avec J. Dieudonné. Éléments de Géométrie Algébrique. Publ. Math. I.H.E.S., 32, 1967.
[2] P. Berthelot. \(\mathcal{ D}\)-modules arithmétiques I. Opérateurs différentiels de niveau fini. Ann. Scient. École. Norm. Sup., \(\mathrm {4}^{{\mathrm e}}\) série, t. 29, pp. 185-272, 1996. · Zbl 0886.14004
[3] W. Fulton. A note on weakly complete algebras. Bull. Amer. Math. Soc., 75, p. 591-593, 1969. · Zbl 0205.34303
[4] C. Huyghe. \(\mathcal{ D}^{\dagger}\)-affinité de l’espace projectif, avec un appendice de P. Berthelot. Compositio Mathematica, 108, No. 3, pp. 277-318, 1997. · Zbl 0956.14010
[5] C. Huyghe. \(\mathcal{ D}^{\dagger}({\scriptstyle\infty})\)-affinité des schémas projectifs. Ann. Inst. Fourier, 48, No. 4, pp. 913-956, 1998. · Zbl 0910.14005
[6] C. Huyghe. Finitude de la dimension cohomologique d’algèbres d’opérateurs différentiels faiblement complètes. Preprint de l’IRMAR, juin 2000.
[7] Z. Mebkhout and L. Narváez-Macarro. Sur les coefficients de de Rham - Grothendieck des variétés algébriques. Proc. Conf. \(p\)-adic Analysis (Trento 1989), Lecture Notes in Math., Springer-Verlag, 1454, pp. 267-308, 1990.
[8] Z. Mebkhout and L. Narváez-Macarro. Sur les coefficients de de Rham - Grothendieck des variétés algébriques II. En préparation, 2000. · Zbl 0727.14011
[9] D. Meredith. Weak formal schemes. Nagoya Math. J., 45, pp. 1-38, 1971. · Zbl 0207.51502
[10] P. Monsky and G. Washnitzer. Formal cohomology I. Annals of Math., 88, pp. 181-217, 1968. · Zbl 0162.52504
[11] A. Virrion. Dualité locale et holonomie pour les \(\mathcal{ D}\)-modules arithmétiques. Bull. Soc. Math. France, t. 128, pp. 101-168, 2000.
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