Abstract :
In this paper we study the homological dimension of skew group rings and crossed products. A sufficient condition for R * G, a crossed product, to have finite right global dimension is given, in terms of crossed products over simple Artinian factors of R if R is right FBN and left coherent and G is finite. Some necessary conditions and sufficient conditions for R * G, a skew group ring of a finite group over a local or semilocal right Noetherian ring, to have finite right global dimension are also given. Then in particular if R is commutative Noetherian and G is finite, we obtain some equivalent conditions for R * G, a skew group ring, to have finite global dimension. Using work of Aljadeff [E. Aljadeff, Serre′s extension theorem for crossed products, J. London Math. Soc.44 (1991), 47-54], these results are extended to polycyclic-by-finite groups.
