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A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula

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Abstract

In this paper we are concerned with absolute, relative and Tate Tor modules. In the first part of the paper we generalize a result of Avramov and Martsinkovsky by using the Auslander—Buchweitz approximation theory, and obtain a new exact sequence connecting absolute Tor modules with relative and Tate Tor modules. In the second part of the paper we consider a depth equality, called the depth formula, which has been initially introduced by Auslander and developed further by Huneke and Wiegand. As an application of our main result, we generalize a result of Yassemi and give a new sufficient condition implying the depth formula to hold for modules of finite Gorenstein and finite injective dimension.

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Acknowledgements

We thank the anonymous referees for many useful comments that helped improve the exposition.

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Authors and Affiliations

  1. Department of Mathematics, West Virginia University, Morgantown, WV, 26506-6310, USA

    Olgur Celikbas

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, P. R. China

    Li Liang

  3. School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou, 730070, P. R. China

    Li Liang

  4. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box: 19395-5746, Tehran, Iran

    Arash Sadeghi & Tirdad Sharif

Authors
  1. Olgur Celikbas
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  2. Li Liang
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  3. Arash Sadeghi
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  4. Tirdad Sharif
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Corresponding author

Correspondence to Li Liang.

Additional information

L. Liang was partly supported by the National Natural Science Foundation of China (Grant Nos. 12271230, 11761045 and 11971388) and the Natural Science Foundation of Gansu Province (Grant No. 21JR7RA297); A. Sadeghi and T. Sharif were partly supported by a grant from IPM

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Celikbas, O., Liang, L., Sadeghi, A. et al. A Study of Tate Homology via the Approximation Theory with Applications to the Depth Formula. Acta. Math. Sin.-English Ser. 39, 439–458 (2023). https://doi.org/10.1007/s10114-023-1190-2

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  • DOI: https://doi.org/10.1007/s10114-023-1190-2

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