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

Abstract

We study the structure of $K$-coalgebras over a field $K$ 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 $K$-coalgebras $C$, 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 $K$, the tame-wild dichotomy holds, for a wide class of coalgebras $C$ of infinite dimension, including the class of semiperfect coalgebras and the class of incidence coalgebras $K^{\square }I$ of interval finite posets $I$. Tools and techniques applied in the study of $K$-coalgebras $C$, 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.