Coalgebras of tame comodule type, comodule categories, and a tame-wild dichotomy problem

  • Daniel Simson

    Nicolaus Copernicus University, Torun, Poland
Coalgebras of  tame comodule  type,  comodule  categories, and a tame-wild dichotomy problem cover
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Abstract

We study the structure of KK-coalgebras over a field KK and comodule categories. In particular, we discuss the concepts of tame comodule type, of discrete comodule type, of polynomial growth, and of wild comodule type, for KK-coalgebras CC, introduced by the author in [83], [84], [93], and intensively studied during the last decade. Among other things, we show that, over an algebraicaly closed field KK, the tame-wild dichotomy holds, for a wide class of coalgebras CC of infinite dimension, including the class of semiperfect coalgebras and the class of incidence coalgebras KIK^{\Box} I of interval finite posets II. Tools and techniques applied in the study of KK-coalgebras CC, their comodules, and representation types, are presented. Characterisations of large classes of coalgebras of tame comodule type are presented, including path coalgebras of quivers, string coalgebras, and the incidence coalgebras of interval finite posets.