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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
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TY - JOUR
T1 - The Ziegler spectrum for derived-discrete algebras
AU - Arnesen, Kristin Krogh
AU - Laking, Rosanna
AU - Pauksztello, David
AU - Prest, Mike
PY - 2017/10/15
Y1 - 2017/10/15
N2 - 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.
AB - 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.
KW - Ziegler spectrum
KW - Pure-injective object
KW - Derived-discrete algebra
KW - Compactly generated triangulated category
KW - Homotopy category
U2 - 10.1016/j.aim.2017.07.016
DO - 10.1016/j.aim.2017.07.016
M3 - Journal article
VL - 319
SP - 653
EP - 698
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
ER -