Tame-wild dichotomy for coalgebras

The concepts of a K-coalgebra C of tame comodule type and of wild comodule type over an algebraically closed field K are introduced in Simson [Colloq. Math. 90 (2001) 101-150] and basic properties of tame coalgebras and wild coalgebras are established in Simson [Colloq. Math. 90 (2001) 101-150; Lect. Notes Pure Appl. Math. 236 (2004) 465-492; J. Pure Appl. Algebra 202 (2005) 118-132; J. Algebra 312 (2007) 455-494]. Unfortunately, the tame-wild dichotomy theorem is proved only for a relatively narrow class of coalgebras. In the present paper, we introduce the concepts fc-tame coalgebra and fc-wild coalgebra, and we prove that any Homcomputable coalgebra over an algebraically closed field K is either fc-tame or fc-wild. Hence we conclude that the usual tame-wild dichotomy holds for semiperfect coalgebras.