# zbMATH — the first resource for mathematics

The relative duality theorem for arithmetic $${\mathcal D}$$-modules. (Théorème de dualité relative pour les $${\mathcal D}$$-modules arithmétiques.) (French) Zbl 0876.14011
Summary: Let $${\mathcal D}$$ be one of the rings of differential operators defined by P. Berthelot [Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 2, 185-272 (1996)] on a smooth scheme of unequal characteristics. We establish that the relative duality theorem for proper morphisms still holds in this context. More precisely we show that the direct image functor for $${\mathcal D}$$-modules commutes with the duality functor [see also A. Virrion, C. R. Acad. Sci., Paris, Sér. I 319, No. 12, 1283-1286 (1994; Zbl 0829.14010)].

##### MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14G20 Local ground fields in algebraic geometry 13N10 Commutative rings of differential operators and their modules