Cosemisimple Coextensions and Homological Dimension of Smash Coproducts

Abstract

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/C̄ is a cosemisimple coextension and C is an injective right C–comodule, then gl. dim(C ⋈ H) = gl. dim(C) = gl. dim (C̄), where gl. dim(C) denotes the global dimension of coalgebra C.