Coalgebras, comodules, pseudocompact algebras and tame comodule type

Daniel Simson

Colloquium Mathematicae, Tome 89 (2001), p. 101-150 / Harvested from The Polish Digital Mathematics Library

Résumé

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 there exists a basic K-coalgebra C and an equivalence of categories ≅ 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.

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Publié le : 2001-01-01

Zbl 1055.16038

EUDML-ID : urn:eudml:doc:283704