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Tate Cohomology for Complexes with Finite Gorenstein AC-Injective Dimension

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. School of Mathematics and Physics, Qingdao University of Science and Technology, Qingdao, 266061, China

    Jianmin Xing

  2. School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong, 273165, China

    Tiwei Zhao

  3. Department of Basic Science, Jinling Institute of Technology, Nanjing, 210093, China

    Yunxia Li

  4. School of Mathematics and Physics, Jiangsu University of Technology, Changzhou, 213001, China

    Jiangsheng Hu

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  1. Jianmin Xing
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  2. Tiwei Zhao
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  3. Yunxia Li
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  4. Jiangsheng Hu
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Correspondence to Tiwei Zhao.

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Communicated by S. Yassemi.

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Xing, J., Zhao, T., Li, Y. et al. Tate Cohomology for Complexes with Finite Gorenstein AC-Injective Dimension. Bull. Iran. Math. Soc. 45, 103–125 (2019). https://doi.org/10.1007/s41980-018-0122-x

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  • DOI: https://doi.org/10.1007/s41980-018-0122-x

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