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Proper resolutions and Gorensteinness in extriangulated categories

Abstract

Let (\(\mathscr{C},\mathbb{E},\frak{s}\)) be an extriangulated category with a proper class ξ of \(\mathbb{E}\)-triangles, and \(\mathscr{W}\) an additive full subcategory of (\(\mathscr{C},\mathbb{E},\frak{s}\)). We provide a method for constructing a proper \(\mathscr{W}\)(ξ)-resolution (resp., coproper \(\mathscr{W}\)(ξ)-coresolution) of one term in an \(\mathbb{E}\)-triangle in ξ from that of the other two terms. By using this way, we establish the stability of the Gorenstein category \(\mathscr{G}\mathscr{W}\)(ξ) in extriangulated categories. These results generalize the work of Z. Y. Huang [J. Algebra, 2013, 393: 142–169] and X. Y. Yang and Z. C. Wang [Rocky Mountain J. Math., 2017, 47: 1013–1053], but the proof is not too far from their case. Finally, we give some applications about our main results.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11771212, 11901190, 11671221), Qing Lan Project of Jiangsu Province, Jiangsu Government Scholarship for Overseas Studies (Grant No. JS-2019-328), Hunan Provincial Natural Science Foundation of China (Grant No. 2018JJ3205), and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 19B239).

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Correspondence to Dondong Zhang.

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Hu, J., Zhang, D. & Zhou, P. Proper resolutions and Gorensteinness in extriangulated categories. Front. Math. China 16, 95–117 (2021). https://doi.org/10.1007/s11464-021-0887-8

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Keywords

  • Proper resolution
  • coproper coresolution
  • extriangulated categories
  • Gorenstein categories

MSC2020

  • 18G80
  • 18E10
  • 18G25
  • 18G10