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Finiteness Conditions and Relative Singularity Categories

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Abstract

We introduce the n-pure projective (resp., injective) dimension of complexes in n-pure derived categories, and give some criteria for computing these dimensions in terms of the n-pure projective (resp., injective) resolutions (resp., coresolutions) and n-pure derived functors. As a consequence, we get some equivalent characterizations for the finiteness of n-pure global dimension of rings. Finally, we study Verdier quotient of bounded n-pure derived category modulo the bounded homotopy category of n-pure projective modules, which is called an n-pure singularity category since it can reflect the finiteness of n-pure global dimension of rings.

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Acknowledgements

The author would like to thank the referees for the very helpful suggestions and corrections that have improved this article.

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Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, P. R. China

    Chun Xia Zhang

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  1. Chun Xia Zhang
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Correspondence to Chun Xia Zhang.

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Supported by National Natural Science Foundation of China (Grant No. 11871125) and Natural Science Foundation of Chongqing (Grant No. cstc2021jcyj-msxmX0048)

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Zhang, C.X. Finiteness Conditions and Relative Singularity Categories. Acta. Math. Sin.-English Ser. 38, 1436–1446 (2022). https://doi.org/10.1007/s10114-022-1072-z

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  • DOI: https://doi.org/10.1007/s10114-022-1072-z

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