Serial coalgebras

AbstractIn this paper we extend the theory of serial and uniserial finite dimensional algebras to coalgebras of arbitrary dimension. Nakayama–Skorniakov Theorems are proved in this new setting and the structure of such coalgebras is determined up to Morita–Takeuchi equivalences. Our main structure theorem asserts that over an algebraically closed field k the basic coalgebra of a serial indecomposable coalgebra is a subcoalgebra of a path coalgebra kΓ where the quiver Γ is either a cycle or a chain (finite or infinite). In the uniserial case, Γ is either a single point or a loop. For cocommutative coalgebras, an explicit description is given, serial coalgebras are uniserial and these are isomorphic to a direct sum of subcoalgebras of the divided power coalgebra