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Zbl 0948.14017
Berthelot, Pierre
${\cal D}$-modules arithmétiques. II: Descente par Frobenius. (Arithmetic ${\cal D}$-modules. II: Frobenius descent). (French)
[J] Mém. Soc. Math. Fr., Nouv. Sér. 81, 138 p. (2000). ISSN 0249-633X

[Part I: {\it P. Berthelot}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 2, 185-272 (1996; Zbl 0886.14004)].\par 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.
[Gerd Faltings (Bonn)]

MSC 2000:: *14F30 p-adic cohomology
14F10 Special sheaves
14F40 De Rham cohomology
16S32 Associative rings of differential operators
32C38 Sheaves of differential operators (analytic spaces)

Keywords: $D$-module; $F$-crystal; $p$-curvature; adjoint operator; rings of differential operators; crystalline cohomology; niveau; Grothendieck-Hartshorne duality; Frobenius-action

Citations: Zbl 0886.14004

Cited in: Zbl 1111.14006 Zbl 1129.14030 Zbl 1056.14025 Zbl 0955.14015

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