Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-29T16:19:45.861Z Has data issue: false hasContentIssue false
  • English
  • Français

Abelian categories arising from cluster tilting subcategories II: quotient functors

Part of: Abelian categories

Published online by Cambridge University Press:  19 July 2019

Yu Liu  and
Panyue Zhou
Yu Liu
Affiliation:
School of Mathematics, Southwest Jiaotong University, 610031Chengdu, Sichuan, People's Republic of China (liuyu86@swjtu.edu.cn)
Panyue Zhou*
Affiliation:
College of Mathematics, Hunan Institute of Science and Technology, 414006Yueyang, Hunan, People's Republic of China (panyuezhou@163.com)
*
*Corresponding author.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we consider a kind of ideal quotient of an extriangulated category such that the ideal is the kernel of a functor from this extriangulated category to an abelian category. We study a condition when the functor is dense and full, in another word, the ideal quotient becomes abelian. Moreover, a new equivalent characterization of cluster tilting subcategories is given by applying homological methods according to this functor. As an application, we show that in a connected 2-Calabi-Yau triangulated category ℬ, a functorially finite, extension closed subcategory 𝒯 of ℬ is cluster tilting if and only if ℬ /𝒯 is an abelian category.

Keywords

Extriangulated categoryabelian categorycluster tilting subcategoryquotient functor

MSC classification

Primary: 18E30: Derived categories, triangulated categories
Secondary: 18E10: Exact categories, abelian categories
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 2721 - 2756
Check for updates
Copyright
Copyright © 2019 The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Demonet, L. and Liu, Y.. Quotients of exact categories by cluster tilting subcategories as module categories. J. Pure Appl. Algebra 217 (2013), 2282–2297.CrossRefGoogle Scholar
2Grimeland, B. and Jacobsen, K.. Abelian quotients of triangulated categories. J. Algebra 439 (2015), 110–133.CrossRefGoogle Scholar
3Koenig, S. and Zhu, B.. From triangulated categories to abelian categories: cluster tilting in a general framework. Math. Z. 258 (2008), 143–160.CrossRefGoogle Scholar
4Liu, Y. and Nakaoka, H.. Hearts of twin cotorsion pairs on extriangulated categories. J. Algebra 528 (2019), 96–149.Google Scholar
5Liu, Y. and Zhou, P.. Abelian categories arising from cluster tilting subcategories. arXiv: 1809.02315v1, 2018.Google Scholar
6Nakaoka, H. and Palu, Y.. Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories. arXiv:1605.05607.Google Scholar
7Zhou, P. and Zhu, B.. Triangulated quotient categories revisited. J. Algebra 502 (2018), 196–232.Google Scholar
8Zhou, P. and Zhu, B.. Cluster-tilting subcategories in extriangulated categories. Theory Appl. Categ. 34 (2019), 221–242.Google Scholar