Relative derived categories, relative singularity categories and relative defect categories. (English) Zbl 1516.18009
Gorenstein homological algebra is a kind of hot relative homological algebra, which is originated from the work of M. Auslander and M. Bridger [Stable module theory. Providence, RI: American Mathematical Society (AMS) (1969; Zbl 0204.36402)]. The main idea of the relative homological algebra is that one might replace projective module by Gorenstein projective modules. The category of Gorenstein projective module is a Frobneius category, by the famous Happel’s Theorem [D. Happel, Comment. Math. Helv. 62, 339–389 (1987; Zbl 0626.16008)], its stable category is a triangulated category. So, the study of Gorenstein homology algebra is combined with the study of triangulated category.
In the present paper, the authors introduce the relative Gorenstein defect category of an abelian category with respect to an admissible subcategory, which generalizing the Gorenstein defect categories in sense of Bergh, Oppermann and Jorgensen [P. A. Bergh et al., Q. J. Math. 66, No. 2, 459–471 (2015; Zbl 1327.13041)]. The authors prove that the relative Gorenstein defect category is triangle equivalent to the relative singularity category with respect to the relative Gorenstein category under a condition. Moreover, the relative Ding projective defect categories are defined and analogous results for it are given in this paper.
In the present paper, the authors introduce the relative Gorenstein defect category of an abelian category with respect to an admissible subcategory, which generalizing the Gorenstein defect categories in sense of Bergh, Oppermann and Jorgensen [P. A. Bergh et al., Q. J. Math. 66, No. 2, 459–471 (2015; Zbl 1327.13041)]. The authors prove that the relative Gorenstein defect category is triangle equivalent to the relative singularity category with respect to the relative Gorenstein category under a condition. Moreover, the relative Ding projective defect categories are defined and analogous results for it are given in this paper.
Reviewer: Zhibing Zhao (Hefei)
MSC:
| 18G80 | Derived categories, triangulated categories |
| 16G50 | Cohen-Macaulay modules in associative algebras |
| 18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |
Keywords:
relative derived category; relative singularity category; relative Gorenstein category; relative Gorenstein defect category; relative Ding projective category; relative Ding defect category.References:
| [1] | J. Asadollahi and R. Hafezi, “On relative Auslander algebras”, Proc. Amer. Math. Soc. 148:6 (2020), 2379-2396. · Zbl 1436.16019 · doi:10.1090/proc/14976 |
| [2] | J. Asadollahi, R. Hafezi, and R. Vahed, “Gorenstein derived equivalences and their invariants”, J. Pure Appl. Algebra 218:5 (2014), 888-903. · Zbl 1291.18013 · doi:10.1016/j.jpaa.2013.10.007 |
| [3] | M. Auslander and M. Bridger, Stable module theory, Memoirs of the American Mathematical Society 94, American Mathematical Society, Providence, RI, 1969. · Zbl 0204.36402 · doi:10.1090/memo/0094 |
| [4] | Y. Bao, X. Du, and Z. Zhao, “Gorenstein singularity categories”, J. Algebra 428 (2015), 122-137. · Zbl 1329.18011 · doi:10.1016/j.jalgebra.2015.01.011 |
| [5] | A. Beligiannis, “The homological theory of contravariantly finite subcategories: Auslander-Buchweitz contexts, Gorenstein categories and (co-)stabilization”, Comm. Algebra 28:10 (2000), 4547-4596. · Zbl 0964.18006 · doi:10.1080/00927870008827105 |
| [6] | P. A. Bergh, S. Oppermann, and D. A. Jorgensen, “The Gorenstein defect category”, Q. J. Math. 66:2 (2015), 459-471. · Zbl 1327.13041 · doi:10.1093/qmath/hav001 |
| [7] | D. Bravo, J. Gillespie, and M. Hovey, “The stable module category of a general ring”, preprint, 2014. |
| [8] | R.-O. Buchweitz, Maximal Cohen-Macaulay modules and Tate cohomology, Mathematical Surveys and Monographs 262, American Mathematical Society, Providence, RI, 2021. · Zbl 1505.13002 · doi:10.1090/surv/262 |
| [9] | T. Cao and W. Ren, “Pure derived and pure singularity categories”, J. Algebra Appl. 19:6 (2020), art. id. 2050117. · Zbl 1440.18021 · doi:10.1142/S0219498820501170 |
| [10] | T. Cao, Z. Liu, and X. Yang, “Derived category with respect to Gorenstein AC-projective modules”, Kodai Math. J. 41:3 (2018), 579-590. · Zbl 1427.18008 · doi:10.2996/kmj/1540951255 |
| [11] | T. Cao, Z. Liu, and X. Yang, “The Gorenstein AC defect category”, Chinese Ann. Math. Ser. A 43:2 (2022), 213-226. · Zbl 1524.18031 · doi:10.16205/j.cnki.cama.2022.0014 |
| [12] | X.-W. Chen, “Homotopy equivalences induced by balanced pairs”, J. Algebra 324:10 (2010), 2718-2731. · Zbl 1239.18010 · doi:10.1016/j.jalgebra.2010.09.002 |
| [13] | X.-W. Chen, “Relative singularity categories and Gorenstein-projective modules”, Math. Nachr. 284:2 (2011), 199-212. · Zbl 1244.18014 · doi:10.1002/mana.200810017 |
| [14] | W. J. Chen, Z. K. Liu, and X. Y. Yang, “Singularity categories with respect to Ding projective modules”, Acta Math. Sin. (Engl. Ser.) 33:6 (2017), 793-806. · Zbl 1387.18019 · doi:10.1007/s10114-017-6209-0 |
| [15] | N. Ding, Y. Li, and L. Mao, “Strongly Gorenstein flat modules”, J. Aust. Math. Soc. 86:3 (2009), 323-338. · Zbl 1200.16010 · doi:10.1017/S1446788708000761 |
| [16] | I. Emmanouil, “On pure acyclic complexes”, J. Algebra 465 (2016), 190-213. · Zbl 1350.16008 · doi:10.1016/j.jalgebra.2016.07.009 |
| [17] | E. E. Enochs and O. M. G. Jenda, “Gorenstein injective and projective modules”, Math. Z. 220:4 (1995), 611-633. · Zbl 0845.16005 · doi:10.1007/BF02572634 |
| [18] | N. Gao and P. Zhang, “Gorenstein derived categories”, J. Algebra 323:7 (2010), 2041-2057. · Zbl 1222.18005 · doi:10.1016/j.jalgebra.2010.01.027 |
| [19] | J. Gillespie, “Model structures on modules over Ding-Chen rings”, Homology Homotopy Appl. 12:1 (2010), 61-73. · Zbl 1231.16005 · doi:10.4310/HHA.2010.v12.n1.a6 |
| [20] | D. Happel, “On the derived category of a finite-dimensional algebra”, Comment. Math. Helv. 62:3 (1987), 339-389. · Zbl 0626.16008 · doi:10.1007/BF02564452 |
| [21] | D. Happel, “On Gorenstein algebras”, pp. 389-404 in Representation theory of finite groups and finite-dimensional algebras (Bielefeld, 1991), edited by G. O. Michler and C. M. Ringel, Progr. Math. 95, Birkhäuser, Basel, 1991. · Zbl 0759.16007 · doi:10.1007/978-3-0348-8658-1_16 |
| [22] | Z. Huang, “Proper resolutions and Gorenstein categories”, J. Algebra 393 (2013), 142-169. · Zbl 1291.18022 · doi:10.1016/j.jalgebra.2013.07.008 |
| [23] | F. Kong and P. Zhang, “From CM-finite to CM-free”, J. Pure Appl. Algebra 220:2 (2016), 782-801. · Zbl 1352.16016 · doi:10.1016/j.jpaa.2015.07.016 |
| [24] | H. Li and Z. Huang, “Relative singularity categories”, J. Pure Appl. Algebra 219:9 (2015), 4090-4104. · Zbl 1417.18006 · doi:10.1016/j.jpaa.2015.02.009 |
| [25] | H. Li and Z. Huang, “Relative singularity categories II”, Kodai Math. J. 43:3 (2020), 431-453. · Zbl 1473.18009 · doi:10.2996/kmj/1605063623 |
| [26] | D. Orlov, “Derived categories of coherent sheaves and triangulated categories of singularities”, pp. 503-531 in Algebra, arithmetic, and geometry, Vol. II, edited by Y. Tschinkel and Y. Zarhin, Progr. Math. 270, Birkhäuser, Boston, MA, 2009. · Zbl 1200.18007 · doi:10.1007/978-0-8176-4747-6_16 |
| [27] | S. Sather-Wagstaff, T. Sharif, and D. White, “Stability of Gorenstein categories”, J. Lond. Math. Soc. (2) 77:2 (2008), 481-502. · Zbl 1140.18010 · doi:10.1112/jlms/jdm124 |
| [28] | L. Tan, D. Wang, and T. Zhao, “Finiteness conditions and relative derived categories”, J. Algebra 573 (2021), 270-296. · Zbl 1471.18009 · doi:10.1016/j.jalgebra.2020.12.034 |
| [29] | J. Šaroch and J. Štovíček, “Singular compactness and definability for \[\Sigma \]-cotorsion and Gorenstein modules”, Selecta Math. (N.S.) 26:2 (2020), art. id. 23. · Zbl 1444.16010 · doi:10.1007/s00029-020-0543-2 |
| [30] | J. P. Wang and Z. X. Di, “Gorenstein regular rings, singularity categories and Ding modules”, Acta Math. Sinica (Chinese Ser.) 62:2 (2019), 331-344. · Zbl 1438.16027 |
| [31] | Z. Wang and Z. Liu, “Stability of strongly Gorenstein flat modules”, Vietnam J. Math. 42:2 (2014), 171-178. · Zbl 1297.16011 · doi:10.1007/s10013-013-0041-3 |
| [32] | J. Wei and J. Dong, “Stability of Ding homological modules”, J. Zhejiang Univ. Sci. Ed. 42:1 (2015), 87-92. · Zbl 1340.16007 |
| [33] | R. Wei, L. Zhongkui, and Y. Gang, “Derived categories with respect to Ding modules”, J. Algebra Appl. 12:6 (2013), art. id. 1350021. · Zbl 1387.18026 · doi:10.1142/S0219498813500217 |
| [34] | X. G. Yan and A. M. Xu, “Stability of Ding projective categories”, J. Nanjing Norm. Univ. Nat. Sci. Ed. 38:1 (2015), 25-31. |
| [35] | G. Yang, Z. Liu, and L. Liang, “Ding projective and Ding injective modules”, Algebra Colloq. 20:4 (2013), 601-612. · Zbl 1280.16003 · doi:10.1142/S1005386713000576 |
| [36] | P. Zhang, “Categorical resolutions of a class of derived categories”, Sci. China Math. 61:2 (2018), 391-402. · Zbl 1396.18011 · doi:10.1007/s11425-016-9117-0 |
| [37] | Y. Zheng and Z. Huang, “On pure derived categories”, J. Algebra 454 (2016), 252-272. · Zbl 1378.16008 · doi:10.1016/j.jalgebra.2016.01.020 |
| [38] | S. J. Zhu, Left homotopy theory and Buchweitz’s theorem, master’s thesis, Shanghai Jiaotong University, 2011 |
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