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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)].

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)