The global dimensions of crossed products and crossed coproducts

In this paper, we show that if H is a finite-dimensional Hopf algebra such that H and H* are semisimple, then gl.dim(A# . H)=gl.dim(A), where . is a convolution invertible cocycle. We also discuss the relationship of global dimensions between the crossed product A# . H and the algebra A, where A is coacted by H. Dually, we give a sufficient condition for a finite dimensional coalgebra C and a finite dimensional semisimple Hopf algebra H such that gl.dim(C . . H)=gl.dim(C).