Gorenstein dimension of abelian categories arising from cluster tilting subcategories

Yu Liu; Panyue Zhou

Czechoslovak Mathematical Journal (2021)

  • Volume: 71, Issue: 2, page 435-453
  • ISSN: 0011-4642

Abstract

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Let 𝒞 be a triangulated category and 𝒳 be a cluster tilting subcategory of 𝒞 . Koenig and Zhu showed that the quotient category 𝒞 / 𝒳 is Gorenstein of Gorenstein dimension at most one. But this is not always true when 𝒞 becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let 𝒞 be an extriangulated category with enough projectives and enough injectives, and 𝒳 a cluster tilting subcategory of 𝒞 . We show that under certain conditions, the quotient category 𝒞 / 𝒳 is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu.

How to cite

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Liu, Yu, and Zhou, Panyue. "Gorenstein dimension of abelian categories arising from cluster tilting subcategories." Czechoslovak Mathematical Journal 71.2 (2021): 435-453. <http://eudml.org/doc/297963>.

@article{Liu2021,
abstract = {Let $\mathcal \{C\}$ be a triangulated category and $\mathcal \{X\}$ be a cluster tilting subcategory of $\mathcal \{C\}$. Koenig and Zhu showed that the quotient category $\mathcal \{C\}/\mathcal \{X\}$ is Gorenstein of Gorenstein dimension at most one. But this is not always true when $\mathcal \{C\}$ becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let $\mathcal \{C\}$ be an extriangulated category with enough projectives and enough injectives, and $\mathcal \{X\}$ a cluster tilting subcategory of $\mathcal \{C\}$. We show that under certain conditions, the quotient category $\mathcal \{C\}/\mathcal \{X\}$ is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu.},
author = {Liu, Yu, Zhou, Panyue},
journal = {Czechoslovak Mathematical Journal},
keywords = {extriangulated category; abelian category; cluster tilting subcategory; Gorenstein dimension},
language = {eng},
number = {2},
pages = {435-453},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Gorenstein dimension of abelian categories arising from cluster tilting subcategories},
url = {http://eudml.org/doc/297963},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Liu, Yu
AU - Zhou, Panyue
TI - Gorenstein dimension of abelian categories arising from cluster tilting subcategories
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 2
SP - 435
EP - 453
AB - Let $\mathcal {C}$ be a triangulated category and $\mathcal {X}$ be a cluster tilting subcategory of $\mathcal {C}$. Koenig and Zhu showed that the quotient category $\mathcal {C}/\mathcal {X}$ is Gorenstein of Gorenstein dimension at most one. But this is not always true when $\mathcal {C}$ becomes an exact category. The notion of an extriangulated category was introduced by Nakaoka and Palu as a simultaneous generalization of exact categories and triangulated categories. Now let $\mathcal {C}$ be an extriangulated category with enough projectives and enough injectives, and $\mathcal {X}$ a cluster tilting subcategory of $\mathcal {C}$. We show that under certain conditions, the quotient category $\mathcal {C}/\mathcal {X}$ is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu.
LA - eng
KW - extriangulated category; abelian category; cluster tilting subcategory; Gorenstein dimension
UR - http://eudml.org/doc/297963
ER -

References

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  1. Demonet, L., Liu, Y., 10.1016/j.jpaa.2013.03.007, J. Pure Appl. Algebra 217 (2013), 2282-2297. (2013) Zbl1408.18021MR3057311DOI10.1016/j.jpaa.2013.03.007
  2. Koenig, S., Zhu, B., 10.1007/s00209-007-0165-9, Math. Z. 258 (2008), 143-160. (2008) Zbl1133.18005MR2350040DOI10.1007/s00209-007-0165-9
  3. Liu, Y., Abelian quotients associated with fully rigid subcategories, Available at https://arxiv.org/abs/1902.07421 (2019), 14 pages. (2019) 
  4. Liu, Y., Nakaoka, H., 10.1016/j.jalgebra.2019.03.005, J. Algebra 528 (2019), 96-149. (2019) Zbl1419.18018MR3928292DOI10.1016/j.jalgebra.2019.03.005
  5. Nakaoka, H., Palu, Y., Extriangulated categories, Hovey twin cotorsion pairs and model structures, Cah. Topol. Géom. Différ. Catég. 60 (2019), 117-193. (2019) Zbl07088229MR3931945
  6. Zhou, P., Zhu, B., 10.1016/j.jalgebra.2018.01.031, J. Algebra 502 (2018), 196-232. (2018) Zbl1388.18014MR3774890DOI10.1016/j.jalgebra.2018.01.031
  7. Zhou, P., Zhu, B., Cluster-tilting subcategories in extriangulated categories, Theory Appl. Categ. 34 (2019), 221-242. (2019) Zbl1408.18029MR3935450

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