Homomorphisms, separable extensions, and Morita maps for weak module algebras

By using a trace one element, we give a sufficient and necessary condition for a weak module algebra $A$ to be a projective left $A\# H$-module, where $A\# H$ denotes the weak smash product. We also give some differentiated conditions for the weak smash product to be a separable extension on the weak module algebra $A$ and get the weak structure theorem in the category of weak $(H,A)$-Hopf modules.