##
**Introduction to the arithmetic theory of \(\mathcal D\)-modules.
(Introduction à la théorie arithmétique des \(\mathcal D\)-modules.)**
*(French)*
Zbl 1098.14010

Berthelot, Pierre (ed.) et al., \(p\)-adic cohomology and arithmetic applications (II). Paris: Société Mathématique de France (ISBN 2-85629-117-1/pbk). Astérisque 279, 1-80 (2002).

The paper under review is the synopsis of a course given by the author at the Centre Émile Borel, France, during the special semester on \(p\)-adic cohomology and its arithmetic applications held in 1997. Its main goal is to provide a comprehensive survey on the recent developments towards a generalization of the classical theory of algebraic \({\mathcal D}\)-modules (over \(\mathbb{C}\)) to the case of base schemes over a field of arbitrary characteristic, emphasizing the arithmetically relevant case of characteristic \(p> 0\).

The objective of such an arithmetic theory of \({\mathcal D}\)-modules is to furnish an adequate toolkit, like in the classical case of characteristic \(0\), for the effective study of the various \(p\)-adic cohomology theories once initiated by A. Grothendieck in the 1960s.

Recently, a remarkable progress in setting-up a \(p\)-adic theory of algebraic \({\mathcal D}\)-modules has been achieved by P. Berthelot himself, mainly through his series of research papers titled “\({\mathcal D}\)-modules arithmétiques I, II, III, IV”, and published between 1996 and 2004.

Actually, the article under review basically surveys the contents of these recent, fundamental papers by explaining systematically both their underlying philosophy and their main results obtained so far.

In this vein, the present treatise is intended as the general introduction to the more detailed research papers “\({\mathcal D}\)-modules arithmétiques I, II, III, IV”, thereby focusing more on the conceptual and methodological framework rather than on complete proofs of the principal results therein. Accordingly, the presentation begins with a sweeping introduction of both historical and strategical nature, which is already very enlightening by itself alone.

The main body of the paper consists of five rather extensive sections treating the following topics:

1. Differential calculus modulo \(p^n\);

2. Cohomology operations modulo \(p^n\);

3. Passage to formal schemes;

4. The characteristic variety and holonomy.

In the course of the entire exposition, the author explains how some fundamental classical results on sheaves of differential operators and their associated sheaves of (coherent) modules can be generalized to the case of characteristic \(p\), and how the construction of different completions can be carried out by using deeper results on formal \(p\)-adic sciemes. Also, the author describes several results and conjectures concerning the concept of holonomy on characteristic \(p> 0\) with regard to \({\mathcal D}\)-modules with Frobenius action.

Altogether, the author has set high value on comprehensive and lucid explanations of the whole matter which, in the abstract, is both conceptually and technically utmost subtle, entangled and advanced. The analogy to the classical prototype on characteristic \(0\) as far as extant, is thoroughly discussed wherever it is appropriate, thereby enforcing the strategical character of this brilliant survey. Without any doubt, this extensive treatise is a perfect introduction to the very recent developments on the arithmetical theory of \({\mathcal D}\)-modules as a whole.

For the entire collection see [Zbl 0990.00020].

The objective of such an arithmetic theory of \({\mathcal D}\)-modules is to furnish an adequate toolkit, like in the classical case of characteristic \(0\), for the effective study of the various \(p\)-adic cohomology theories once initiated by A. Grothendieck in the 1960s.

Recently, a remarkable progress in setting-up a \(p\)-adic theory of algebraic \({\mathcal D}\)-modules has been achieved by P. Berthelot himself, mainly through his series of research papers titled “\({\mathcal D}\)-modules arithmétiques I, II, III, IV”, and published between 1996 and 2004.

Actually, the article under review basically surveys the contents of these recent, fundamental papers by explaining systematically both their underlying philosophy and their main results obtained so far.

In this vein, the present treatise is intended as the general introduction to the more detailed research papers “\({\mathcal D}\)-modules arithmétiques I, II, III, IV”, thereby focusing more on the conceptual and methodological framework rather than on complete proofs of the principal results therein. Accordingly, the presentation begins with a sweeping introduction of both historical and strategical nature, which is already very enlightening by itself alone.

The main body of the paper consists of five rather extensive sections treating the following topics:

1. Differential calculus modulo \(p^n\);

2. Cohomology operations modulo \(p^n\);

3. Passage to formal schemes;

4. The characteristic variety and holonomy.

In the course of the entire exposition, the author explains how some fundamental classical results on sheaves of differential operators and their associated sheaves of (coherent) modules can be generalized to the case of characteristic \(p\), and how the construction of different completions can be carried out by using deeper results on formal \(p\)-adic sciemes. Also, the author describes several results and conjectures concerning the concept of holonomy on characteristic \(p> 0\) with regard to \({\mathcal D}\)-modules with Frobenius action.

Altogether, the author has set high value on comprehensive and lucid explanations of the whole matter which, in the abstract, is both conceptually and technically utmost subtle, entangled and advanced. The analogy to the classical prototype on characteristic \(0\) as far as extant, is thoroughly discussed wherever it is appropriate, thereby enforcing the strategical character of this brilliant survey. Without any doubt, this extensive treatise is a perfect introduction to the very recent developments on the arithmetical theory of \({\mathcal D}\)-modules as a whole.

For the entire collection see [Zbl 0990.00020].

Reviewer: Werner Kleinert (Berlin)

### MSC:

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14F40 | de Rham cohomology and algebraic geometry |

14G22 | Rigid analytic geometry |

16S32 | Rings of differential operators (associative algebraic aspects) |

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

13N10 | Commutative rings of differential operators and their modules |

14D15 | Formal methods and deformations in algebraic geometry |