| Title:
|
Gorenstein dimension of abelian categories arising from cluster tilting subcategories (English) |
| Author:
|
Liu, Yu |
| Author:
|
Zhou, Panyue |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
71 |
| Issue:
|
2 |
| Year:
|
2021 |
| Pages:
|
435-453 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $\mathscr {C}$ be a triangulated category and $\mathscr {X}$ be a cluster tilting subcategory of $\mathscr {C}$. Koenig and Zhu showed that the quotient category $\mathscr {C}/\mathscr {X}$ is Gorenstein of Gorenstein dimension at most one. But this is not always true when $\mathscr {C}$ 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 $\mathscr {C}$ be an extriangulated category with enough projectives and enough injectives, and $\mathscr {X}$ a cluster tilting subcategory of $\mathscr {C}$. We show that under certain conditions, the quotient category $\mathscr {C}/\mathscr {X}$ is Gorenstein of Gorenstein dimension at most one. As an application, this result generalizes the work by Koenig and Zhu. (English) |
| Keyword:
|
extriangulated category |
| Keyword:
|
abelian category |
| Keyword:
|
cluster tilting subcategory |
| Keyword:
|
Gorenstein dimension |
| MSC:
|
18E10 |
| MSC:
|
18G80 |
| idZBL:
|
07361078 |
| idMR:
|
MR4263179 |
| DOI:
|
10.21136/CMJ.2021.0417-19 |
| . |
| Date available:
|
2021-05-20T13:43:11Z |
| Last updated:
|
2023-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148914 |
| . |
| Reference:
|
[1] Demonet, L., Liu, Y.: Quotients of exact categories by cluster tilting subcategories as module categories.J. Pure Appl. Algebra 217 (2013), 2282-2297. Zbl 1408.18021, MR 3057311, 10.1016/j.jpaa.2013.03.007 |
| Reference:
|
[2] Koenig, S., Zhu, B.: From triangulated categories to abelian categories: Cluster tilting in a general framework.Math. Z. 258 (2008), 143-160. Zbl 1133.18005, MR 2350040, 10.1007/s00209-007-0165-9 |
| Reference:
|
[3] Liu, Y.: Abelian quotients associated with fully rigid subcategories.Available at https://arxiv.org/abs/1902.07421 (2019), 14 pages. |
| Reference:
|
[4] Liu, Y., Nakaoka, H.: Hearts of twin cotorsion pairs on extriangulated categories.J. Algebra 528 (2019), 96-149. Zbl 1419.18018, MR 3928292, 10.1016/j.jalgebra.2019.03.005 |
| Reference:
|
[5] Nakaoka, H., Palu, Y.: Extriangulated categories, Hovey twin cotorsion pairs and model structures.Cah. Topol. Géom. Différ. Catég. 60 (2019), 117-193. Zbl 07088229, MR 3931945 |
| Reference:
|
[6] Zhou, P., Zhu, B.: Triangulated quotient categories revisited.J. Algebra 502 (2018), 196-232. Zbl 1388.18014, MR 3774890, 10.1016/j.jalgebra.2018.01.031 |
| Reference:
|
[7] Zhou, P., Zhu, B.: Cluster-tilting subcategories in extriangulated categories.Theory Appl. Categ. 34 (2019), 221-242. Zbl 1408.18029, MR 3935450 |
| . |
