Stability of Gorenstein Subcategories in Extriangulated Length Categories

WU Yuchen; LIU Dajun; WANG Chuanyu; LIU Dahong

doi:10.1051/wujns/2026315448

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Stability of Gorenstein Subcategories in Extriangulated Length Categories

Mathematics | 更新时间：2026-10-29

- Stability of Gorenstein Subcategories in Extriangulated Length Categories
- Wuhan University Journal of Natural Sciences Vol. 31, Issue 5, Pages: 448-458(2026)
- 作者机构：
  
  1.安徽工程大学数理与金融学院，安徽芜湖 241000
  2.皖西中学，安徽六安 237000
- 作者简介：
  
  WU Yuchen, female, Master candidate, research direction: homological algebra. E-mail: 2325386199@qq.com
  E-mail: liudajun@ahpu.edu.cn
- 基金信息：
- DOI：10.1051/wujns/2026315448
  CLC： O154.2
- Received：27 November 2025，
  
  Published：24 October 2026
- 稿件说明：
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吴雨晨,刘大俊,王传玉等.外三角长度范畴中Gorenstein子范畴的稳定性[J].武汉大学学报自然科学英文版,2026,31(05):448-458.

WU Yuchen, LIU Dajun, WANG Chuanyu, et al. Stability of Gorenstein Subcategories in Extriangulated Length Categories[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 448-458.
吴雨晨,刘大俊,王传玉等.外三角长度范畴中Gorenstein子范畴的稳定性[J].武汉大学学报自然科学英文版,2026,31(05):448-458. DOI： 10.1051/wujns/2026315448.

WU Yuchen, LIU Dajun, WANG Chuanyu, et al. Stability of Gorenstein Subcategories in Extriangulated Length Categories[J]. Wuhan Univ J of Nat Sci, 2026, 31(5): 448-458. DOI： 10.1051/wujns/2026315448.

摘要

设

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279082&type=

1.97555566

2.68111110

是一个长度函数，

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</mo><mi>θ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279123&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279097&type=

16.08666801

3.52777815

为具有足够多

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279099&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279085&type=

1.97555566

2.68111110

-投射对象的外三角长度范畴。我们引入了

</mo><mi mathvariant="double-struck">E</mi><mo>

</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo><mo>

</mo><mi>θ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279123&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279097&type=

16.08666801

3.52777815

中Gorenstein子范畴

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279087&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279111&type=

9.03111172

3.52777815

和

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279113&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279101&type=

10.72444439

3.80999994

的定义，并且证明了对于任意的

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279104&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279128&type=

7.05555630

2.68111110

，有

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279117&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279105&type=

22.86000061

3.80999994

。该结论刻画了外三角长度范畴中的Gorenstein同调性质。

Abstract

Let

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279027&type=

1.69333339

2.25777793

be a length function and

</mo><mi mathvariant="double-struck">E</mi><mo>

</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo><mo>

</mo><mi>θ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279066&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279040&type=

13.82888889

2.96333337

be an extriangulated length category with enough

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279042&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279067&type=

1.69333339

2.25777793

-projective objects. We introduce the concepts of the Gorenstein subcategories

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279031&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279043&type=

7.76111078

2.96333337

and

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279071&type=

9.31333351

3.24555564

</mo><mi mathvariant="double-struck">E</mi><mo>

</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo><mo>

</mo><mi>θ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279073&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279046&type=

13.82888889

2.96333337

and we prove

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279060&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279047&type=

19.61444283

3.24555564

for any

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https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279075&type=

6.06777763

2.25777793

. This result provides a characterization of the Gorenstein homological property in

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</mo><mi mathvariant="normal">s</mi><mo stretchy="false">)</mo><mo>

</mo><mi>θ</mi><mo stretchy="false">)</mo></math>

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279073&type=

https://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=121279046&type=

13.82888889

2.96333337

关键词

Keywords

references

Auslander M , Bridger M . Stable module theory [M ] . Providence : American Mathematical Society , 1969 .

Enochs E E , Jenda O M G . Gorenstein injective and projective modules [J ] . Mathematische Zeitschrift , 1995 , 220 ( 1 ): 611 - 633 .

Sather-Wagstaff K A , Sharif T , White D . Stability of Gorenstein categories [J ] . Journal of the London Mathematical Society , 2008 , 77 ( 2 ): 481 - 502 .

Huang Z Y . Proper resolutions and Gorenstein categories [J ] . Journal of Algebra , 2013 , 393 : 142 - 169 .

Yang X Y , Wang Z C . Proper resolutions and Gorensteinness in triangulated categories [J ] . Rocky Mountain Journal of Mathematics , 2017 , 47 ( 3 ): 1013 - 1053 .

Nakaoka H , Palu Y . Extriangulated categories, Hovey twin cotorsion pairs and model structures [J ] . Cahiers de Topologie et Géométrie Différentielle Catégoriques , 2019 , 60 ( 2 ): 117 - 193 .

Hu J S , Zhang D D , Zhou P Y . Proper classes and Gorensteinness in extriangulated categories [J ] . Journal of Algebra , 2020 , 551 : 23 - 60 .

Hu J S , Zhang D D , Zhou P Y . Proper resolutions and Gorensteinness in extriangulated categories [J ] . Frontiers of Mathematics in China , 2021 , 16 ( 1 ): 95 - 117 .

Hu J S , Zhang D D , Zhou P Y . Gorenstein homological dimensions for extriangulated categories [J ] . Bulletin of the Malaysian Mathematical Sciences Society , 2021 , 44 ( 4 ): 2235 - 2252 .

Hu J S , Lu J F , Zhang D D , et al . Stable categories and triangulated equivalences [J ] . Scientia Sinica Mathematica , 2023 , 53 ( 4 ): 565 - 576 (Ch) .

He J , Zhou P Y . Homological systems in extriangulated categories [J ] . Scientia Sinica Mathematica , 2023 , 53 ( 5 ): 697 - 716 (Ch) .

Wang L , Wei J Q , Zhang H C , et al . Extriangulated length categories: Torsion classes and τ -tilting theory [EB/OL ] . [ 2025-10-30 ] . https://arxiv.org/abs/2502.07367 https://arxiv.org/abs/2502.07367 .

Zheng Y L , Zhang P Y . Gorenstein projective objects in extriangulated length categories [EB/OL ] . [ 2026-09-30 ] . https://chinaxiv.org/abs/202610.00015(Ch) https://chinaxiv.org/abs/202610.00015(Ch) .

Bühler T . Exact categories [J ] . Expositiones Mathematicae , 2010 , 28 ( 1 ): 1 - 69 .

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