HOMOLOGICAL DIMENSION OF SKEW GROUP-RINGS AND CROSSED-PRODUCTS

In this paper we study the homological dimension of skew group rings a nd crossed products. A sufficient condition for R G, a crossed produ ct, to have finite right global dimension is given, in terms of crosse d products over simple Artinian factors of R if R is right FBN and lef t coherent and G is finite. Some necessary conditions and sufficient c onditions for R G, a skew group ring of a finite group over a local or semilocal right Noetherian ring, to have finite right global dimens ion are also given. Then in particular if R is commutative Noetherian and G is finite, we obtain some equivalent conditions for R G, a ske w group ring, to have finite global dimension. Using work of Aljadeff [E. Aljadeff, Serre's extension theorem for crossed products, J. Londo n Math. Soc. 44 (1991), 47-54], these results are extended to polycycl ic-by-finite groups. (C) 1994 Academic Press, Inc.