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A basic course on differential modules. (English) Zbl 0853.32011
Maisonobe, Philippe (ed.) et al., $$D$$-modules cohérents et holonomes. Cours d’été du CIMPA ’Éléments de la théorie des systèmes différentiels’, août et septembre 1990, Nice, France. Paris: Hermann. Trav. Cours. 45, 103-168 (1993).
This paper is a very good introduction to the theory of $$\mathcal D$$-modules: modules over the (sheaf of) ring(s) $$\mathcal D$$ of linear differential operators on a complex analytic variety. Chapter I is devoted to the definition and first properties of the ring $$\mathcal D$$, first of all on $$\mathbb{C}^n$$ and then on a complex analytic variety $$X$$. The filtration by the order is introduced and, by using Oka’s theorem ($${\mathcal O}_X$$ is coherent) the coherence of $${\mathcal D}_X$$ is proved. The chapter ends with some results on the Weyl algebra on a field of characteristic zero. Chapter II is centered on the study of filtrations and good filtrations on (left) $$\mathcal D$$-modules. (In proposition 12, part 1, the hypothesis $$\mathcal M$$ is $$\mathcal D$$-coherent is missing and necessary). Locally good filtrations are characterized in terms of the coherence of the associated graded module. Chapter III deals with the main geometric object associated to a coherent $$\mathcal D$$-module $$\mathcal M$$: its characteristic variety, $$\text{Char} ({\mathcal M})$$. Bernstein’s inequality is proved. A complete (microlocal) proof of the involutivity of the characteristic variety is given in the appendix. This proof is due to Malgrange and simplifies the proof he gave at the “Séminaire Bourbaki” [B. Malgrange, Lect. Notes Math. 710, 277-289 (1979; Zbl 0423.46033)]. In chapter IV the general theory of holonomic $$\mathcal D$$-modules is developed. Coherent $$\mathcal D$$-modules with characteristic variety equal to $$T^*_X X$$ (i.e. the zero section of $$T^*X$$ the cotangent bundle to $$X$$) are characterized as locally free (of finite rank) $${\mathcal O}$$-modules endowed with a holomorphic integrable connection. As an application of this result it is proved that if $$\mathcal M$$ is $$\mathcal D$$-coherent and $$\text{Char}({\mathcal M}) = T^*_Y X$$ (i.e. the conormal bundle to a smooth hypersurface $$Y \subset X$$) then $$\mathcal M$$ is locally isomorphic to a direct sum of a finite number of copies of $${\mathcal O}[*Y]/{\mathcal O}$$ (here $${\mathcal O}[*Y]$$ denotes the sheaf of meromorphic functions with poles along $$Y$$). The chapter ends with a proof of the existence of a minimal polynomial for an endomorphism of a holonomic $$\mathcal D$$-module. Dimension and multiplicity (at a point of $$T^*X)$$ of a coherent $$\mathcal D$$-module, are studied in chapter V. In particular their behavior in exact sequences and, as an application, the proof that, for a holonomic $$\mathcal D$$-module $$\mathcal M$$, the $${\mathcal D}_x$$-module $${\mathcal M}_x$$ is of finite length, for all $$x \in X$$. The chapter ends with some results on good graded resolutions for coherent $$\mathcal D$$-modules, the homological dimension of the fibers of $$\mathcal D$$ (which is equal to $$\dim X$$) and the existence of a canonical filtration for each coherent $$\mathcal D$$-module. Chapter VI is an Epilogue. The authors give a proof (inspired by M. Kashiwara [ Invent. Math. 38, 33-53 (1976; Zbl 0354.35082)]) of the existence of the Bernstein-Sato polynomial associated to a germ of an analytic function $$f$$ on $$X$$. As an application they give a proof of the coherence and the holonomicity of $${\mathcal O}[1/f]$$ (the sheaf of meromorphic functions with poles along $$f = 0$$).
For the entire collection see [Zbl 0824.00033].

##### MSC:
 32C38 Sheaves of differential operators and their modules, $$D$$-modules