Picard groups and strongly graded coalgebras

In this paper we study strongly graded coalgebras and its relation to the Picard group. A classification theorem for this kind of coalgebras is given via the second Doiʹs cohomology group. The strong Picard group of a coalgebra is introduced in order to characterize those graded coalgebras with strongly graded dual ring. Finally, for a Hopf algebra H we also characterize the H*-Galois coextensions with dual H-Galois extension solving the question proposed in Dimagescimagelescu et al., J. Algebra 178 (1995) 400–413.