On the homogenized Weyl Algebra

The aim of this paper is to give relations between the category of finetely generated graded modules over the homogeneized Weyl algebra $B_{n}$, the finetely generated modules over the Weyl algebra $A_{n}$ and the finetely generated graded modules over the Yoneda algebra $B_{n}^{!}$ of $B_{n}$. We will give these relations both at the level of the categories of modules and at the level of the derived categories.