Proper resolutions and Gorensteinness in extriangulated categories

doi:10.1007/s11464-021-0887-8

Front. Math. China

2021, Vol. 16

Issue (1) : 95-117 https://doi.org/10.1007/s11464-021-0887-8

RESEARCH ARTICLE

Proper resolutions and Gorensteinness in extriangulated categories

Jiangsheng HU¹, Dondong ZHANG²(

), Panyue ZHOU³

¹. Department of Mathematics, Jiangsu University of Technology, Changzhou 213001, China
². Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China
³. College of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

Download: PDF(317 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

Let $(ℓ, E, s)$ be an extriangulated category with a proper class $ξ$ of $E$ -triangles, and $W$ an additive full subcategory of $(ℓ, E, s)$ . We provide a method for constructing a proper $W ξ$ -resolution (resp., coproper $W ξ$ - coresolution) of one term in an $E$ -triangle in $ξ$ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category $G W ξ$ in extriangulated categories. These results generalize the work of Z. Y. Huang [J. Algebra, 2013, 393: 142{169] and X. Y. Yang and Z. C. Wang [Rocky Mountain J. Math., 2017, 47: 1013{1053], but the proof is not too far from their case. Finally, we give some applications about our main results.

Keywords Proper resolution coproper coresolution extriangulated categories Gorenstein categories

Corresponding Author(s): Dondong ZHANG

Issue Date: 26 March 2021

Cite this article:

Jiangsheng HU,Dondong ZHANG,Panyue ZHOU. Proper resolutions and Gorensteinness in extriangulated categories[J]. Front. Math. China, 2021, 16(1): 95-117.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-021-0887-8
https://academic.hep.com.cn/fmc/EN/Y2021/V16/I1/95

1	A Beligiannis. Relative homological algebra and purity in triangulated categories. J Algebra, 2000, 227: 268–361 https://doi.org/10.1006/jabr.1999.8237
2	J S Hu, D D Zhang, P Y Zhou. Proper classes and Gorensteinness in extriangulated categories. J Algebra, 2020, 551: 23–60 https://doi.org/10.1016/j.jalgebra.2019.12.028
3	Z Y Huang. Proper resolutions and Gorenstein categories. J Algebra, 2013, 393: 142–169 https://doi.org/10.1016/j.jalgebra.2013.07.008
4	X Ma, T W Zhao, Z Y Huang. Resolving subcategories of triangulated categories and relative homological dimension. Acta Math Sinica, 2017, 33: 1513–1535 https://doi.org/10.1007/s10114-017-6416-8
5	N Nakaoka, Y Palu. Extriangulated categories, Hovey twin cotorsion pairs and model structures. Cah Topol Géom Différ Catég, 2019, 60(2): 117–193
6	W Ren, Z K Liu. Gorenstein homological dimensions for triangulated categories. J Algebra, 2014, 410: 258–276 https://doi.org/10.1016/j.jalgebra.2014.03.037
7	S Sather-Wagstaff, T, Sharif D. WhiteStability of Gorenstein categories. J Lond Math Soc, 2008, 77: 481–502 https://doi.org/10.1112/jlms/jdm124
8	Z P, Wang S T Guo, C L Liang. Stability of Gorenstein objects in triangulated categories. Sci Sin Math (Chin Ser), 2017, 47: 349–356(in Chinese) https://doi.org/10.1360/012016-3
9	X Y Yang. Notes on proper class of triangles. Acta Math Sinica (Engl Ser), 2013, 29: 2137–2154 https://doi.org/10.1007/s10114-013-2435-2
10	X Y Yang. Model structures on triangulated categories. Glasg Math J, 2015, 57: 263–284 https://doi.org/10.1017/S0017089514000299
11	X Y Yang, Z C Wang. Proper resolutions and Gorensteiness in triangulated categories. Rocky Mountain J Math, 2017, 47: 1013–1053 https://doi.org/10.1216/RMJ-2017-47-3-1013
12	P Y Zhou, B Zhu. Triangulated quotient categories revisited. J Algebra, 2018, 502: 196–232 https://doi.org/10.1016/j.jalgebra.2018.01.031

[1]	Li LIANG. Vanishing of stable homology with respect to a semidualizing module[J]. Front. Math. China, 2018, 13(1): 107-127.

Viewed

Full text

Abstract

Cited

Shared

Discussed