Coalgebras, comodules, pseudocompact algebras and tame comodule type

• Autorzy: Daniel Simson
• Języki publikacji: EN

Abstrakty

EN

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 𝔄 there exists a basic K-coalgebra C and an equivalence of categories 𝔄 ≅ C-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 $_{C}Q*$ opposite to the Gabriel quiver $_{C}Q$ of C is a pure semisimple locally Dynkin quiver (see Section 9) and C is isomorphic to the path K-coalgebra $K(_{C}Q)$. Open questions are formulated in Section 10.

Kategorie tematyczne

• 16G20: Representations of quivers and partially ordered sets
• 16W80: Topological and ordered rings and modules
• 16G60: Representation type (finite, tame, wild, etc.)
• 16G70: Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2001
• Tom: 90
• Numer: 1
• Strony: 101-150

Daty

wydano

2001

Twórcy

autor

Daniel Simson

• Faculty of Mathematics and Computer Science, Nicholas Copernicus University, Chopina 12/18, 87-100 Toruń, Poland

Identyfikatory

DOI

10.4064/cm90-1-9