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$${\mathcal D}$$-modules arithmétiques. II: Descente par Frobenius. (Arithmetic $${\mathcal D}$$-modules. II: Frobenius descent). (French) Zbl 0948.14017
[Part I: P. Berthelot, Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 2, 185-272 (1996; Zbl 0886.14004)].
The paper under review is another step in the author’s systematic study of rings of differential operators in crystalline cohomology. These are filtered by the niveau which measures what type of factorials appear in the denominator. The main result is that Frobenius raises the niveau by 1 and induces (up to that change of niveau) an equivalence of categories of $$D$$-modules. This is preceded by a comparison of left and right $$D$$-modules, which are exchanged by Grothendieck-Hartshorne duality. Finally it is shown that Frobenius commutes with the usual (“six”) operations, and applications to Frobenius-action on cohomology and $$F$$-$$D$$-modules are given.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14F40 de Rham cohomology and algebraic geometry 16S32 Rings of differential operators (associative algebraic aspects) 32C38 Sheaves of differential operators and their modules, $$D$$-modules
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