Let H be a finite dimensional cocommutative Hopf algebra and A an H-module algebra. In this paper, we characterize the projectivity (injectivity) of M as a left A#σH-module when it is projective (injective) as a left A-module. The sufficient and necessary condition for A#σH, the crossed product, to have finite global homological dimension is given, in terms of the global homological dimension of A and the surjectivity of trace maps, provided that H is cocommutative and A is commutative.