## Biduality theorem and characterisation of holonomic $$F$$-$${\mathcal D}_{{\mathcal X},\mathbb{Q}}^ \dagger$$-modules. (Théorème de bidualité et caractérisation des $$F$$-$${\mathcal D}_{{\mathcal X},\mathbb{Q}}^ \dagger$$-modules holonomes.)(French)Zbl 0829.14010

The author gives:
(1) a biduality theorem for the perfect complexes of left (or right) modules over the ring of differential operators $${\mathcal D}^†_X$$, introduced by P. Berthelot [in: $$p$$-adic Analysis, Proc. Int. Conf., Trento 1989, Lect. Notes Math. 1454, 80-124 (1990; Zbl 0722.14008)], for $$X$$ a smooth scheme over a field of characteristic $$p>0$$, or a smooth formal scheme over a complete principal valuation ring of characteristic $$(0,p)$$ [see also Z. Mebkhout and the reviewer, ibid. 267-308 (1990; Zbl 0727.14011) for the case of weakly formal schemes];
(2) the commutation of the duality functor with the inverse image functor by the Frobenius morphism; and
(3) a homological characterization of holonomicity for the coherent $${\mathcal D}_{{\mathcal X}, Q}^†$$-modules equipped with a Frobenius structure [P. Berthelot, “$${\mathcal D}_Q^†$$-modules cohérents, II, III” (in preparation)], where $${\mathcal X}$$ is a smooth formal scheme over the Witt ring of a perfect field of characteristic $$p$$.
Points (1) and (3) are conceptually similar to the corresponding results in the classic cases of complex smooth analytic varieties or smooth algebraic varieties over a field of characteristic 0. Key points in the proof are the structure of $${\mathcal D}^†_X$$ as a direct limit of the rings of differential operators of finite level, and the fact that the dualizing sheaf $$\omega_X$$ has a canonical structure of right $${\mathcal D}_X^†$$-module [P. Berthelot (loc. cit.); see also B. Haastert, Manuscr. Math. 62, No. 3, 341-354 (1988; Zbl 0673.14012) and Z. Mebkhout and the reviewer (loc. cit., 1.1.4)].

### MSC:

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

### Keywords:

Frobenius morphism; ring of differential operators

### Citations:

Zbl 0722.14008; Zbl 0727.14011; Zbl 0673.14012