ℰ X -modules

This chapter is devoted to the study of the algebraic structure of the sheaf of rings ℰ_X and of the usual operations on Modules over ℰ_X. The Ring ℰ_X has a Z-filtration induced by the order, and many properties of a ℤ-filtered Module M over a ℤ-filtered Ring A can be obtained from the corresponding ones on gr(M), the associated graded Module over gr(A), if we make the assumption that the filtration is what we call a “zariskian filtration”. We study the theory of filtered modules in the abstract in § 1, with special attention to characteristic ideals, flatness, homological dimension, coherency. The important theorem of O. Gabber [1] on the involutivity of the characteristic ideal is stated, but it seemed irrelevant to us to duplicate his very clear proof.Next we have to show that the filtration over ℰ_X is in fact a zariskian one. The proof follows that of Sato-Kashiwara-Kawaï [1] and is classical in its approach: induction on the dimension with the help of the division theorems. The main properties of ℰ_X then follow immediately: the sheaf ℰ_X is coherent and noetherian, and the characteristic cycle of a coherent ℰ_X-module is a conic analytic involutive cycle. As a corollary of the involutivity we get that the homological dimension of ℰ_X is equal to n the (complex) dimension of X, and even that any coherent Module locally admits a free resolution of length at most n.It is beyond the scope of this book to make a detailed study of holonomic Modules (i.e.: coherent Modules whose characteristic variety is Lagrangean). We just give the first properties of the category of holonomic Modules, and study the “adjoint” functor on this category.We end § 2 by introducing the “dummy variable” trick which enables to work outside of the zero section of T*X, and by finally studying the relations between coherent D_X-modules and globally defined coherent ℰ_X-modules.Next we explain how one manipulates coherent gℰ_X-modules with the same ease as is possible for functions: tensor products for two Modules over two different manifolds, inverse images for left ℰ_X-modules (analogous to restriction for functions), direct images of right Modules (in line with integration of differential forms). All these constructions have been done in Sato-Kashiwara-Kawaï [1] using a deep cohomological method, but we proceed here in an elementary way, beginning with the study of these operations for Modules like I_X|X, then passing to the general case. Non-characteristic restrictions of ℰ_X-modules are studied in detail, and we calculate exactly, following M. Kashiwara [5], the characteristic cycle of the induced system in this case.On the other hand direct images for proper maps are not studied here, and reference is made to Kashiwara [3] and Pham [1] for further developments on this subject (cf. also Chapter III, § 4).Finally we give a general formulation of the Cauchy problem for general systems, as preparation for the next chapter.