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).

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).

Reviewer: W.Kleinert (Berlin)

##### 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 |