On the homological dimension of group algebras of solvable groups

In the paper we calculate the weak dimension of the group algebra of a solvable group and the projective dimension of the group algebra of a countable nilpotent group. Exact bounds are obtained for the projective dimension of the group algebra of a torsion-free solvable group. For the case that the principal ring is commutative and Noetherian we obtain corresponding results for global dimensions. On the assumption that the principal ring is a field we prove the additivity of projective and weak dimension on a class of groups for which there exists a finitely generated projective resolvent of the principal ring.