The Ziegler spectrum for derived-discrete algebras
The Ziegler spectrum for derived-discrete algebras
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DOI：
10.1016/j.aim.2017.07.016
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发表时间：
2016-03
影响因子：
1.7
通讯作者：
K. Arnesen;Rosanna Laking;David Pauksztello;M. Prest
中科院分区：
文献类型：
--
作者：
K. Arnesen;Rosanna Laking;David Pauksztello;M. Prest
设Λ是一个导离散代数。本文证明了投射Λ-模的同伦范畴的Krull-Gabriel维数为2，从而证明了投射Λ-模的齐格勒谱的Cantor-Bendixson秩为2，推广了Bobienski和Krause [8]的一个结果.我们还描述了所有不可分解的纯内射复形，从而得到了导离散代数的齐格勒谱，推广了Z. Han [17].利用这一点，我们证明了投射Λ-模的同伦范畴中的所有不可分解复形都是纯内射的，从而得到了一类代数，对于这类代数，每个不可分解复形都是纯内射的，但它们不是纯半单的。
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.