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Relative Cohomology and Generalized Tate Cohomology

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Abstract

We compare relative cohomology theories arising from using different proper resolutions of modules. Criteria for the vanishing of such distinctions are given in certain cases, and we show that this is related to the generalized Tate cohomology theory. We also demonstrate that the two balance properties admitted by the two different cohomology theories are actually equivalent in some cases. As applications, we recover many results obtained earlier in various contexts. At last we investigate derived functors with respect to the Auslander and Bass classes.

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Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (Grant No. 11571165). The authors would like to give many thanks to the referee for many useful comments and suggestions.

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Bin Yu

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, China

    Xiaosheng Zhu & Yanbo Zhou

Authors
  1. Bin Yu
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  2. Xiaosheng Zhu
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  3. Yanbo Zhou
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Correspondence to Bin Yu.

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Presented by Henning Krause.

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Yu, B., Zhu, X. & Zhou, Y. Relative Cohomology and Generalized Tate Cohomology. Algebr Represent Theor 20, 1571–1592 (2017). https://doi.org/10.1007/s10468-017-9702-3

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  • DOI: https://doi.org/10.1007/s10468-017-9702-3

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