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Tame‐wild dichotomy for coalgebras
Author(s) -
Simson Daniel
Publication year - 2008
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn047
Subject(s) - coalgebra , algebraically closed field , field (mathematics) , type (biology) , pure mathematics , mathematics , algebra over a field , class (philosophy) , computer science , biology , ecology , artificial intelligence
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.