Coalgebras, comodules, pseudocompact algebras and tame comodule type

Daniel 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 -