Homological dimension of coalgebras and crossed coproducts

Let H be a Hopf k-algebra. We study the global homological dimension of the underlying coalgebra structure of H. We show that gl.dim(H) is equal to the injective dimension of the trivial right H-comodule k. We also prove that if D = C x(alpha)H is a crossed coproduct with invertible alpha, then gl.dim(D) less than or equal to gl.dim(C) + gl.dim(H). Some applications of this result are obtained. Moreover, if C is a cocommutative coalgebra such that C* is noetherian, then the global dimension of the coalgebra C coincides with the global dimension of the algebra C*.