Cotorsion dimensions and Hopf algebra actions

Let H be a finite-dimensional Hopf algebra over a field k, and let A be an H-module algebra. In this paper, we discuss the cotorsion dimension of the smash product A # H. We prove that $$l.\cot .D\left( {A\# H} \right) \leqslant l.\cot .D\left( A \right) + r.D\left( H \right),$$ which generalizes the result of group rings. Moreover, we give some sufficient conditions for which $$l.\cot .D\left( {A\# H} \right) = l.\cot .D\left( A \right).$$ As applications, we study the invariants of IF properties and Gorenstein global dimensions.