Coalgebras, comodules, pseudocompact algebras and tame comodule type
Volume 90 / 2001
Abstract
We develop a technique for the study of $K$-coalgebras and their representation types by applying a quiver technique and topologically pseudocompact modules over pseudocompact $K$-algebras in the sense of Gabriel [17], [19]. A definition of tame comodule type and wild comodule type for $K$-coalgebras over an algebraically closed field $K$ is introduced. Tame and wild coalgebras are studied by means of their finite-dimensional subcoalgebras. A weak version of the tame-wild dichotomy theorem of Drozd [13] is proved for a class of $K$-coalgebras. By applying [17] and [19] it is shown that for any length $K$-category ${\frak A}$ there exists a basic $K$-coalgebra $C$ and an equivalence of categories ${\frak A}\cong C\hbox{-}{\rm comod}$. This allows us to define tame representation type and wild representation type for any abelian length $K$-category.
Hereditary coalgebras and path coalgebras $KQ$ of quivers $Q$ are investigated. Tame path coalgebras $KQ$ are completely described in Theorem 9.4 and the following $K$-coalgebra analogue of Gabriel's theorem [18] is established in Theorem 9.3. An indecomposable basic hereditary $K$-coalgebra $C$ is left pure semisimple (that is, every left $C$-comodule is a direct sum of finite-dimensional $C$-comodules) if and only if the quiver $_CQ^*$ opposite to the Gabriel quiver ${}_CQ$ of $C$ is a pure semisimple locally Dynkin quiver (see Section 9) and $C$ is isomorphic to the path $K$-coalgebra $K(_CQ)$. Open questions are formulated in Section~10.
