### Abstract

Let Γ be a coalgebra over a field k. We introduce an operator Tr that takes a right quasi-finitely copresented Γ-comodule M to a left quasi-finitely copresented Γ-comodule Tr M. If M is indecomposable not injective and Tr M is finite-dimensional over K, we prove the existence of an almost split sequence 0 → M → E → DTr M → 0 in the category of all right Γ-comodules, where D = Hom_{k}( , k). If Γ is right semiperfect and the embedding of each simple right comodule S into its injective envelope I(S) has the property that the socle of I(S)/S is finite-dimensional, the above almost split sequence exists for each finite-dimensional M, and DTr M is also finite-dimensional.

Original language | English (US) |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Journal of Algebra |

Volume | 249 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 2002 |

### Keywords

- Almost split sequence
- Coalgebra
- Comodule
- Semiperfect
- Transpose

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

*Journal of Algebra*,

*249*(1), 1-19. https://doi.org/10.1006/jabr.2001.9086