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Tame-wild dichotomy for coalgebras

2008; Wiley; Volume: 78; Issue: 3 Linguagem: Inglês

10.1112/jlms/jdn047

1469-7750

Daniel Simson,

Rings, Modules, and Algebras

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.