Cosemisimple Coextensions and Homological Dimension of Smash Coproducts

Let H be a finite dimensional cosemisimple Hopf algebra, C a left H.comodule coalgebra and let C . = C/C (H*)+be the quotient coalgebra and C . H the smash coproduct. It is shown that if C/C. is a cosemisimple coextension and C is an injective right C.comodule, then gl. dim(C . H) = gl. dim(C) = gl. dim (C.), where gl. dim(C) denotes the global dimension of coalgebra C.