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The Ziegler spectrum for derived-discrete algebras

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The Ziegler spectrum for derived-discrete algebras. / Arnesen, Kristin Krogh; Laking, Rosanna; Pauksztello, David et al.
In: Advances in Mathematics, Vol. 319, 15.10.2017, p. 653-698.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Arnesen, KK, Laking, R, Pauksztello, D & Prest, M 2017, 'The Ziegler spectrum for derived-discrete algebras', Advances in Mathematics, vol. 319, pp. 653-698. https://doi.org/10.1016/j.aim.2017.07.016

APA

Arnesen, K. K., Laking, R., Pauksztello, D., & Prest, M. (2017). The Ziegler spectrum for derived-discrete algebras. Advances in Mathematics, 319, 653-698. https://doi.org/10.1016/j.aim.2017.07.016

Vancouver

Arnesen KK, Laking R, Pauksztello D, Prest M. The Ziegler spectrum for derived-discrete algebras. Advances in Mathematics. 2017 Oct 15;319:653-698. Epub 2017 Sept 11. doi: 10.1016/j.aim.2017.07.016

Author

Arnesen, Kristin Krogh ; Laking, Rosanna ; Pauksztello, David et al. / The Ziegler spectrum for derived-discrete algebras. In: Advances in Mathematics. 2017 ; Vol. 319. pp. 653-698.

Bibtex

@article{e765e650464d4a3d89b86425c6818753,
title = "The Ziegler spectrum for derived-discrete algebras",
abstract = "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 Bobi{\'n}ski 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.",
keywords = "Ziegler spectrum, Pure-injective object, Derived-discrete algebra, Compactly generated triangulated category, Homotopy category",
author = "Arnesen, {Kristin Krogh} and Rosanna Laking and David Pauksztello and Mike Prest",
year = "2017",
month = oct,
day = "15",
doi = "10.1016/j.aim.2017.07.016",
language = "English",
volume = "319",
pages = "653--698",
journal = "Advances in Mathematics",
issn = "0001-8708",
publisher = "Academic Press Inc.",

}

RIS

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 -