Bulletin of the Iranian Mathematical Society

, Volume 45, Issue 1, pp 103–125 | Cite as

Tate Cohomology for Complexes with Finite Gorenstein AC-Injective Dimension

  • Jianmin Xing
  • Tiwei ZhaoEmail author
  • Yunxia Li
  • Jiangsheng Hu
Original Paper


In this paper, we introduce and study a notion of Gorenstein AC-injective dimension for complexes of left modules over associative rings. We show first that the class of complexes with finite Gorenstein AC-injective dimension is exactly the class of complexes admitting a complete \(\mathcal {AC}\)-coresolution. Then the interaction between the corresponding relative and Tate cohomologies of complexes is given. Finally, the relationships between Gorenstein AC-injective dimensions and injective dimensions for complexes are given.


Complexes Absolutely clean Gorenstein AC-injective Tate cohomology 

Mathematics Subject Classification

Primary 18G25 Secondary 16E30 16E10 



This research was partially supported by National Natural Science Foundation of China (11501257,11771212,11671069, 11571164), China Postdoctoral Science Foundation funded project (2016M600426), Postgraduate Research and Innovation Program of Jiangsu Province (KYZZ_160034), Research Project of Teaching Reform in Undergraduate Colleges and Universities in Shandong Province 2016(Z2016Z005), Nanjing University Innovation and Creative Program for PhD candidate (2016011) and Jinling Institute of Technology of China (jit-b-201638, jit-fhxm-2-1707), Qing Lan Project of Jiangsu Province. The authors would like to thank the referee for many considerable suggestions and showing us Corollary 3.13 and Remark 3.14, which have greatly improved this paper.


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Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  • Jianmin Xing
    • 1
  • Tiwei Zhao
    • 2
    Email author
  • Yunxia Li
    • 3
  • Jiangsheng Hu
    • 4
  1. 1.School of Mathematics and PhysicsQingdao University of Science and TechnologyQingdaoChina
  2. 2.School of Mathematical SciencesQufu Normal UniversityQufuChina
  3. 3.Department of Basic ScienceJinling Institute of TechnologyNanjingChina
  4. 4.School of Mathematics and PhysicsJiangsu University of TechnologyChangzhouChina

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