Czechoslovak Mathematical Journal, Vol. 69, No. 1, pp. 117-129, 2019


Gorenstein projective complexes with respect to cotorsion pairs

Renyu Zhao, Pengju Ma

Received April 25, 2017.   Published online July 10, 2018.

Abstract:  Let $(\mathcal{A,B})$ be a complete and hereditary cotorsion pair in the category of left $R$-modules. In this paper, the so-called Gorenstein projective complexes with respect to the cotorsion pair $(\mathcal{A}, \mathcal{B})$ are introduced. We show that these complexes are just the complexes of Gorenstein projective modules with respect to the cotorsion pair $(\mathcal{A}, \mathcal{B})$. As an application, we prove that both the Gorenstein projective modules with respect to cotorsion pairs and the Gorenstein projective complexes with respect to cotorsion pairs possess stability.
Keywords:  cotorsion pair; Gorenstein projective complex with respect to cotorsion pairs; stability of Gorenstein categories
Classification MSC:  18G25, 18G35
DOI:  10.21136/CMJ.2018.0194-17

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] S. Bouchiba: Stability of Gorenstein classes of modules. Algebra Colloq. 20 (2013), 623-636. DOI 10.1142/S100538671300059X | MR 3116791 | Zbl 1281.16010
[2] S. Bouchiba, M. Khaloui: Stability of Gorenstein flat modules. Glasg. Math. J. 54 (2012), 169-175. DOI 10.1017/S0017089511000516 | MR 2862395 | Zbl 1235.16009
[3] D. Bravo, J. Gillespie: Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44 (2016), 2213-2233. DOI 10.1080/00927872.2015.1044100 | MR 3490674 | Zbl 1346.18021
[4] E. E. Enochs, J. R. García Rozas: Gorenstein injective and projective complexes. Commun. Algebra 26 (1998), 1657-1674. DOI 10.1080/00927879808826229 | MR 1622438 | Zbl 0908.18007
[5] E. E. Enochs, O. M. G. Jenda: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. DOI 10.1007/BF02572634 | MR 1363858 | Zbl 0845.16005
[6] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[7] J. R. García Rozas: Covers and Envelopes in the Category of Complexes of Modules. Chapman & Hall/CRC Research Notes in Mathematics 407, Chapman & Hall/CRC, Boca Raton (1999). MR 1693036 | Zbl 0922.16001
[8] J. Gillespie: The flat model structure on Ch($R$). Trans. Am. Math. Soc. 356 (2004), 3369-3390. DOI 10.1090/S0002-9947-04-03416-6 | MR 2052954 | Zbl 1056.55011
[9] H. Holm: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. DOI 10.1016/j.jpaa.2003.11.007 | MR 2038564 | Zbl 1050.16003
[10] J. S. Hu, A. M. Xu: On stability of F-Gorenstein flat categories. Algebra Colloq. 23 (2016), 251-262. DOI 10.1142/S1005386716000286 | MR 3475049 | Zbl 1346.16006
[11] L. Liang, N. Q. Ding, G. Yang: Some remarks on projective generators and injective cogenerators. Acta Math. Sin., Engl. Ser. 30 (2014), 2063-2078. DOI 10.1007/s10114-014-3227-z | MR 3285935 | Zbl 1304.18032
[12] S. Sather-Wagstaff, T. Sharif, D. White: Stability of Gorenstein categories. J. Lond. Math. Soc., II. Ser. 77 (2008), 481-502. DOI 10.1112/jlms/jdm124 | MR 2400403 | Zbl 1140.18010
[13] A. M. Xu, N. Q. Ding: On stability of Gorenstein categories. Commun. Algebra 42 (2013), 3793-3804. DOI 10.1080/00927872.2012.677892 | MR 3169490 | Zbl 1284.16006
[14] G. Yang, Z. K. Liu: Cotorsion pairs and model structure on Ch($R$). Proc. Edinb. Math. Soc., II. Ser. 54 (2011), 783-797. DOI 10.1017/S0013091510000489 | MR 2837480 | Zbl 1238.13023
[15] G. Yang, Z. K.Liu: Stability of Gorenstein flat categories. Glasg. Math. J. 54 (2012), 177-191. DOI 10.1017/S0017089511000528 | MR 2862396 | Zbl 1248.16007
[16] X. Y. Yang, W. J. Chen: Relative homological dimensions and Tate cohomology of complexes with respect to cotorsion pairs. Commun. Algebra 45 (2017), 2875-2888. DOI 10.1080/00927872.2016.1233226 | MR 3594565 | Zbl 1372.18015
[17] X. Y. Yang, N. Q. Ding: On a question of Gillespie. Forum Math. 27 (2015), 3205-3231. DOI 10.1515/forum-2013-6014 | MR 3420339 | Zbl 1347.18003
[18] X. Y. Yang, Z. K. Liu: Gorenstein projective, injective, and flat complexes. Commun. Algebra 39 (2011), 1705-1721. DOI 10.1080/00927871003741497 | MR 2821502 | Zbl 1238.16002

Affiliations:   Renyu Zhao (corresponding author), Department of Mathematics, Northwest Normal University, Anning East road No. 967, Lanzhou, Gansu, 730070, P. R. China, and Department of Mathematics, Nanjing University, Nanjing, 210093, P. R. China, e-mail: zhaory@nwnu.edu.cn; Pengju Ma, Department of Mathematics, Northwest Normal University, Anning East road No. 967, Lanzhou, Gansu, 730070, P. R. China, e-mail: 2642293920@qq.com


 
PDF available at: