The Ziegler spectrum for derived-discrete algebras

Arnesen, Kristin Krogh and Laking, Rosanna and Pauksztello, David and Prest, Mike (2017) The Ziegler spectrum for derived-discrete algebras. Advances in Mathematics, 319. pp. 653-698. ISSN 0001-8708

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Let Λ be a derived-discrete algebra. We show that the Krull–Gabriel dimension of the homotopy category of projective Λ-modules, and therefore the Cantor–Bendixson rank of its Ziegler spectrum, is 2, thus extending a result of Bobiński 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 Λ-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.

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Journal Article
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Advances in Mathematics
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18 Sep 2017 15:28
Last Modified:
19 Sep 2023 01:47