# Coalgebras, comodules, pseudocompact algebras and tame comodule type

Colloquium Mathematicae (2001)

- Volume: 90, Issue: 1, page 101-150
- ISSN: 0010-1354

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topDaniel Simson. "Coalgebras, comodules, pseudocompact algebras and tame comodule type." Colloquium Mathematicae 90.1 (2001): 101-150. <http://eudml.org/doc/283704>.

@article{DanielSimson2001,

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 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.},

author = {Daniel Simson},

journal = {Colloquium Mathematicae},

keywords = {tame representation type; wild coalgebras; finite-dimensional subcoalgebras; topologically pseudocompact modules; pseudocompact algebras; wild representation type; path coalgebras of quivers},

language = {eng},

number = {1},

pages = {101-150},

title = {Coalgebras, comodules, pseudocompact algebras and tame comodule type},

url = {http://eudml.org/doc/283704},

volume = {90},

year = {2001},

}

TY - JOUR

AU - Daniel Simson

TI - Coalgebras, comodules, pseudocompact algebras and tame comodule type

JO - Colloquium Mathematicae

PY - 2001

VL - 90

IS - 1

SP - 101

EP - 150

AB - 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.

LA - eng

KW - tame representation type; wild coalgebras; finite-dimensional subcoalgebras; topologically pseudocompact modules; pseudocompact algebras; wild representation type; path coalgebras of quivers

UR - http://eudml.org/doc/283704

ER -

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