Acknowledgement
The authors sincerely thank the referee for the helpful suggestions and valuable comments. This work was supported by National Natural Science Foundation of China (Nos. 11761060, 11901463), Science and Technology Project of Gansu Province (20JR5RA517), Innovation Ability Enhancement Project of Gansu Higher Education Institutions (2019A-002) and Improvement of Young Teachers' Scientific Research Ability (NWNU-LKQN-18-30).
References
- V. Becerril, O. Mendoza, M. A. Perez, and V. Santiago, Frobenius pairs in abelian categories. Correspondences with cotorsion pairs, exact model categories, and Auslander-Buchweitz contexts, J. Homotopy Relat. Struct. 14 (2019), no. 1, 1-50. https://doi.org/10.1007/s40062-018-0208-4
- A. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, in Analysis and topology on singular spaces, I (Luminy, 1981), 5-171, Asterisque, 100, Soc. Math. France, Paris, 1982.
- D. Bravo, J. Gillespie, and M. Hovey, The stable module category of a general ring, arXiv:1405.5768, 2014.
- T. Buhler, Exact categories, Expo. Math. 28 (2010), no. 1, 1-69. https://doi.org/10.1016/j.exmath.2009.04.004
- W. Chen, Z. Liu, and X. Yang, A new method to construct model structures from a cotorsion pair, Comm. Algebra 47 (2019), no. 11, 4420-4431. https://doi.org/10.1080/00927872.2018.1527922
- N. Ding, Y. Li, and L. Mao, Strongly Gorenstein flat modules, J. Aust. Math. Soc. 86 (2009), no. 3, 323-338. https://doi.org/10.1017/S1446788708000761
- E. E. Enochs and O. M. G. Jenda, Gorenstein injective and projective modules, Math. Z. 220 (1995), no. 4, 611-633. https://doi.org/10.1007/BF02572634
- E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, De Gruyter Expositions in Mathematics, 30, Walter de Gruyter & Co., Berlin, 2000. https://doi.org/10.1515/9783110803662
- E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra. Volume 2, De Gruyter Expositions in Mathematics, 54, Walter de Gruyter GmbH & Co. KG, Berlin, 2011.
- J. Gillespie, Model structures on exact categories, J. Pure Appl. Algebra 215 (2011), no. 12, 2892-2902. https://doi.org/10.1016/j.jpaa.2011.04.010
- J. Gillespie, How to construct a Hovey triple from two cotorsion pairs, Fund. Math. 230 (2015), no. 3, 281-289. https://doi.org/10.4064/fm230-3-4
- J. Gillespie, Gorenstein complexes and recollements from cotorsion pairs, Adv. Math. 291 (2016), 859-911. https://doi.org/10.1016/j.aim.2016.01.004
- J. Gillespie, Duality pairs and stable module categories, J. Pure Appl. Algebra 223 (2019), no. 8, 3425-3435. https://doi.org/10.1016/j.jpaa.2018.11.010
- J. Gillespie and A. Iacob, Duality pairs, generalized Gorenstein modules, and Ding injective envelopes, C. R. Math. Acad. Sci. Paris 360 (2022), 381-398. https://doi.org/10.5802/crmath.306
- R. Gobel and J. Trlifaj, Approximations and Endomorphism Algebras of Modules. Volume 2, second revised and extended edition, De Gruyter Expositions in Mathematics, 41, Walter de Gruyter GmbH & Co. KG, Berlin, 2012. https://doi.org/10.1515/9783110218114
- H. Holm and P. Jorgensen, Cotorsion pairs induced by duality pairs, J. Commut. Algebra 1 (2009), no. 4, 621-633. https://doi.org/10.1216/JCA-2009-1-4-621
- M. Hovey, Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), no. 3, 553-592. https://doi.org/10.1007/s00209-002-0431-9
- B. Keller, Derived categories and their uses, in Handbook of algebra, Vol. 1, 671-701, Handb. Algebr., 1, Elsevier/North-Holland, Amsterdam, 1996. https://doi.org/10.1016/S1570-7954(96)80023-4
- Z. Wang, G. Yang, and R. Zhu, Gorenstein flat modules with respect to duality pairs, Comm. Algebra 47 (2019), no. 12, 4989-5006. https://doi.org/10.1080/00927872.2019.1609011
- X. Yang and W. Chen, Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs, Comm. Algebra 45 (2017), no. 7, 2875-2888. https://doi.org/10.1080/00927872.2016.1233226