Graded coalgebras and Morita-Takeuchi contexts

Viewing a G-graded k-coalgebra over the field $k$ as a right $k$ G-comodule coalgebra it is possible to use a Hopf algebraic approach to the study of coalgebras graded by an arbitrary group that was started in [NT]. Let $C=\oplus_{g\in G}C_{g}$ be a G-graded coalgebra. The graded C-comodules may be viewed as comodules over the smash product $C\rangle\triangleleft kG$ , the general definition of which was given in [M]. Coalgebras graded by an arbitrary group have been considered in [FM] in order to lntroduce the notion of G-graded Hopf algebras. On the other hand, M. Takeuchi introduced in [T] the sets of preequivalence data connecting categories of comodules over two coalgebras (we call such a set a Morita-Takeuchi context). The main result of this note is a coalgebra version of a result established by M. Cohen, S. Montgomery in [CM] for group-graded rings: for a graded coalgebra $C$ the coalgebras $C_{1}$ and $C\rangle\not\in kG$ are connected by a Morita-Takeuchi context in which one of the structure maps is injective. Most of the results in this note are consequences of the foregoing. As a first application we find that a coalgebra $C$ is strongly graded if and only if the other structure map of the context is also injective. The final section provides analogues of the Cohen-Montgomery duality theorems: if $C$ is a coalgebra graded by the flnite group $G$ of order $n$ , then $G$ acts on the smash coproduct as a group of automorphisms of coalgebras and $(C\rangle\triangleleft kG)*kG^{*}$ is coalgebra isomorphic to the comatrix coalgebra $M^{c}(n, C)$ . If $G$ is a finite group of order $n$ , acting on the coalgebra $D$ as a group of coalgebra automorphisms, then the smash coproduct $D\rangle\triangleleft kG^{*}$ is strongly graded by $G$ and moreover: