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Cohomologie rigide et théorie des $${\mathcal D}$$-modules. (Rigid cohomology and theory of $${\mathcal D}$$-modules). (French) Zbl 0722.14008
p-adic analysis, Proc. Int. Conf., Trento/Italy 1989, Lect. Notes Math. 1454, 80-124 (1990).
[For the entire collection see Zbl 0707.00010.]
Let k a perfect field of characteristic p, W the ring of Witt vectors over k, and K the quotient field of W. Due to the author’s previous results [Mém. Soc. Math. Fr., Nouv. Sér. 23, 7-32 (1986; Zbl 0606.14017)], the rigid cohomology groups $$H^ i(X/K)$$, associated with any separated k-scheme X of finite type, coincide with the crystalline cohomology groups if X is proper and smooth, and with the Monsky- Washnitzer cohomology groups if X is affine and smooth. On the other hand, one may assign to X a certain category of coefficients, the so- called category of surconvergent F-isocrystals, which leads to an interpretation of the Dwork cohomology as a rigid cohomology with coefficients in an isocrystal.
In the paper under review, the author studies the problem of establishing Grothendieck’s “formalism of the six operations” for F-isocrystals. This formalism is well-established for coherent sheaves and étale sheaves in étale cohomology theory, and it serves as a basic tool for treating Grothendieck’s problem concerning the coefficients in the De Rham cohomology of schemes [cf. A. Grothendieck in: Dix exposés cohomologie schémes, Adv. Stud. Pure Math. 3, 306-359 (1968; Zbl 0215.371)]. In the recent decade, Grothendieck’s program concerning the De Rham coefficients has successfully been worked out for the groundfield $${\mathbb{C}}$$ of complex numbers, thanks to the work of Kashiwara and Mebkhout about the theory of holonomic $${\mathcal D}$$-modules and the Riemann-Hilbert correspondence.
As in the p-adic case the De Rham cohomology and the rigid cohomology are linked by certain relations, the author tries to carry over the usual theory of $${\mathcal D}$$-modules to formal schemes over discrete valuation rings with residue field k $$(char(k)=p)$$, and to link the corresponding $${\mathcal D}$$-modules to the rigid cohomology. This ambitious, very substantial and technically difficult program is worked out in five chapters, and that in a very detailed manner. The main results consist in constructing a suitable sheaf of differential operators on a formal smooth scheme, describing its action on different types of module sheaves arising from the rigid cohomology, studying the corresponding $${\mathcal D}$$- modules with respect to coherence, investigating the De Rham complexes of those $${\mathcal D}$$-modules, and examining various types of surconvergent isocrystals by means of local cohomology of $${\mathcal D}$$-modules.
As for an even more systematic and detailed account on his p-adic theory of $${\mathcal D}$$-modules, the author refers to his forthcoming work entitled “$${\mathcal D}^+$$-modules cohérents” (in preparation).

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 14G20 Local ground fields in algebraic geometry 14F40 de Rham cohomology and algebraic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies
##### Citations:
Zbl 0707.00010; Zbl 0606.14017; Zbl 0215.371