The role of gentle algebras in higher homological algebra

Johanne Haugland; Karin M. Jacobsen; Sibylle Schroll

doi:10.1515/forum-2021-0311

The role of gentle algebras in higher homological algebra

Johanne Haugland, Karin M. Jacobsen^*, Sibylle Schroll

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

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Abstract

We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra Λ contains a d-cluster tilting subcategory for some d ≥ 2, then Λ is a radical square zero Nakayama algebra. This gives a complete classification of weakly d-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if D^b(Λ) contains a d-cluster tilting subcategory that is closed under [d], then Λ is derived equivalent to an algebra of Dynkin type A. Furthermore, our approach gives a geometric characterization of all d-cluster tilting subcategories of D^b(Λ) that are closed under [d].

Original language	English
Journal	Forum Mathematicum
Volume	34
Issue	5
Pages (from-to)	1255-1275
Number of pages	21
ISSN	0933-7741
DOIs	https://doi.org/10.1515/forum-2021-0311
Publication status	Published - Sept 2022

Keywords

(d + 2)-angulated category
d-abelian category
d-cluster tilting subcategory
Gentle algebra
higher homological algebra

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10.1515/forum-2021-0311

https://arxiv.org/pdf/2107.01045.pdf

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title = "The role of gentle algebras in higher homological algebra",

abstract = "We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra Λ contains a d-cluster tilting subcategory for some d ≥ 2, then Λ is a radical square zero Nakayama algebra. This gives a complete classification of weakly d-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if Db(Λ) contains a d-cluster tilting subcategory that is closed under [d], then Λ is derived equivalent to an algebra of Dynkin type A. Furthermore, our approach gives a geometric characterization of all d-cluster tilting subcategories of Db(Λ) that are closed under [d].",

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The role of gentle algebras in higher homological algebra

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