## Abstract

Let be a derived-discrete algebra. We show that the Krull-Gabriel dimension

of the homotopy category of projective -modules, and therefore the Cantor-

Bendixson rank of its Ziegler spectrum, is 2, thus extending a result of Bobinski and Krause [8]. We also describe all the indecomposable pure-injective complexes and hence the Ziegler spectrum for derived-discrete algebras, extending a result of Z. Han [17]. Using this, we are able to prove that all indecomposable complexes in the homotopy category of projective -modules are pure-injective, so obtaining a class of algebras for which every indecomposable complex is pure-injective but which are not derived pure-semisimple.

of the homotopy category of projective -modules, and therefore the Cantor-

Bendixson rank of its Ziegler spectrum, is 2, thus extending a result of Bobinski and Krause [8]. We also describe all the indecomposable pure-injective complexes and hence the Ziegler spectrum for derived-discrete algebras, extending a result of Z. Han [17]. Using this, we are able to prove that all indecomposable complexes in the homotopy category of projective -modules are pure-injective, so obtaining a class of algebras for which every indecomposable complex is pure-injective but which are not derived pure-semisimple.

Original language | English |
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Pages (from-to) | 653-698 |

Journal | Advances in Mathematics |

Volume | 319 |

Issue number | 0 |

DOIs | |

Publication status | Published - 11 Sept 2017 |