Ɗ-Modules

In this Chapter we will introduce the notion of a Ɗ-module and we will provide its first fundamental properties. We will then also microlocalize such a notion, as to introduce the notion of ℰ-module. Since homological algebra is an important underlying notion for the theory of Ɗ-modules, in this introduction we will provide a concise and self-contained review of the theory of derived categories and of spectral sequences for a double complex. In section 2, we discuss the foundations for Ɗ-modules in the spirit of algebraic geometry. In section 3, we introduce the notion of a good filtration on Ɗ-module which allows us to define the characteristic variety of a Ɗ-module. In particular, we prove that such a characteristic variety depends only upon the module structure and not on a particularly chosen good filtration. In the last section of this Chapter we prove Sato’s fundamental theorem for the category of ℰ-modules.