Let $H$ be a finite dimensional hopf algebra and $B \subset A$ be an $H$-extension of algebras. We know that the following are equivelant
1) $A \cong B \times_\sigma H$ is a cocycle crossed product of $B$ with $H$ for some cocyle $\sigma: H \otimes H \to C$
2) $B \subset A$ is a Cleft $H$-Extension.
3) $A$ $H$-Galois and $A$ is $B$-free and if $\{f_i\}$ is a dual basis for $H^\ast$, then there exist a $u\in A$ such that $\{f_i \cdot u\}$ is a $B$-basis for $A$.
Now we have a dual notion for 1), that is, a cocycle crossed coproduct coalgebra. That is, if $C$ is a coalgebra, and $H$ coacts on $C$ by $\beta: H \otimes C$, $\beta(c) = \sum c^{(1)}\otimes c^{(2)}$, and we have a cocylce $\psi: C \to H \otimes H$ $\psi(c) = \sum \psi(c)^{(1)}\otimes\psi(c)^{(2)}$, then we have a new coalgebra $C^\psi\times H$ built from $C \otimes H$ with coproduct $\Delta(c\otimes h) = \sum c_{(1)}\otimes c_{(2)}^{(1)}\psi(c_{(3)})^{(1)}h_{(1)}\otimes c_{(2)}^{(2)} \otimes \psi(c_{(3)})^{(2)}h_{(2)}$
What are the dual notions for 2 and 3? are they also equivalent?
