  G-algebras, or Groebner bases algebras, were considered by Levandovsky, these
algebras include very important families of algebras, like the Weyl algebras
and the universal enveloping algebra of a finite dimensional Lie algebra. These
algebras are not graded in a natural way, but they are filtered. Going to the
associated graded algebra is the standard way to study their properties. In a
previous paper we studied an homogeneous version of the G-algebras, which
results naturally graded, and through a des homogenization property, we obtain
an ordinary G-algebra. The main results in our first paper were the following:
Homogeneous G-algebras are Koszul of finite global dimension, Artin Schelter
regular, noetherian and have a Poincare Birkoff basis. We give the structure of
their Yoneda algebras by generators and relations. In the second part of the
paper we consider the quantum polynomial ring and prove, for the finitely
generated graded modules, cohomology formulas analogous to those we have in the
usual polynomial ring. In the third part we use the results of the previous
part to study the cohomology of the homogeneous G-algebras. Part four was
dedicated to the study of the relations between an homogeneous G-algebra B and
its des homogenization B/(Z-1)B=A, Applications to the study of finite
dimensional Lie algebras and Weyl algebras were given. In the last sections of
the paper we study the relations, at the level of modules, among the
homogenized algebra B, its graded localization B_{z}, the des homogenization
A=B/(Z-1)B and the Yoneda algebra B^{!}. In this paper we investigate the
relations among all these algebras at the level of derived categories.
