Rings of differential operators on singular varieties

This project will be centered in the study of rings of differential operators on singular varieties and the theory of D-modules. We will start with the ring of k-linear differential operators on a polynomial ring. We will prove that it is a simple Noetherian domain and compute its dimension. Also, considering modules over this ring we will show Bernstein s inequality that give us a lower bound to the dimension of the module and, using this result, we will define holonomic D-modules. Our goal is to study what good structural properties of the regular case can be extended in the singular case. We will use Weyl algebras to give a description of the ring of k-linear differential operators on a finitely generated k-algebra. We will show that they do not have as much as good properties as Weyl algebras, however we will obtain some of them under several conditions. We will study modules over these rings and their dimension, proving a generalized Bernstein s inequality under some conditions. Finally, we will study the case of the ring of differential operators on a hyperplane arrangement and explain several methods to obtain a system of generators