Morita, Galois, and the trace map: a survey

Let A be a K -algebra and H a K -bialgebra ( K being a field). Any action β of H on A gives rise to two new K -algebras, namely, the algebra A β of the invariants of A under β and the smash product A # β H , as well as a canonical Morita context connecting them. Such a context keeps a close relation with the notion of Galois extension. Indeed, in some cases where it makes sense the strictness of this context is equivalent to exactly say that A is a H ∗ -Galois extension of A β . In general, such an equivalence depends also on the surjectivity of a certain trace map from A to A β . This paper is a survey about the strictness of this context in the setting of partial actions of groups and of Hopf algebras.