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Keywords:
cotorsion pair; Gorenstein projective complex with respect to cotorsion pairs; stability of Gorenstein categories
cotorsion pair; Gorenstein projective complex with respect to cotorsion pairs; stability of Gorenstein categories
Summary:
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.
References:
[1] Bouchiba, S.: Stability of Gorenstein classes of modules. Algebra Colloq. 20 (2013), 623-636. DOI 10.1142/S100538671300059X | MR 3116791 | Zbl 1281.16010
[2] Bouchiba, S., Khaloui, M.: Stability of Gorenstein flat modules. Glasg. Math. J. 54 (2012), 169-175. DOI 10.1017/S0017089511000516 | MR 2862395 | Zbl 1235.16009
[3] Bravo, D., Gillespie, J.: 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] Enochs, E. E., Rozas, J. R. García: Gorenstein injective and projective complexes. Commun. Algebra 26 (1998), 1657-1674. DOI 10.1080/00927879808826229 | MR 1622438 | Zbl 0908.18007
[5] Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. DOI 10.1007/BF02572634 | MR 1363858 | Zbl 0845.16005
[6] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). DOI 10.1515/9783110803662 | MR 1753146 | Zbl 0952.13001
[7] Rozas, J. R. García: 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] Gillespie, J.: 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] Holm, H.: 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] Hu, J. S., Xu, A. M.: On stability of F-Gorenstein flat categories. Algebra Colloq. 23 (2016), 251-262. DOI 10.1142/S1005386716000286 | MR 3475049 | Zbl 1346.16006
[11] Liang, L., Ding, N. Q., Yang, G.: 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] Sather-Wagstaff, S., Sharif, T., White, D.: 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] Xu, A. M., Ding, N. Q.: On stability of Gorenstein categories. Commun. Algebra 42 (2013), 3793-3804. DOI 10.1080/00927872.2012.677892 | MR 3169490 | Zbl 1284.16006
[14] Yang, G., Liu, Z. K.: 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] Yang, G., K.Liu, Z.: Stability of Gorenstein flat categories. Glasg. Math. J. 54 (2012), 177-191. DOI 10.1017/S0017089511000528 | MR 2862396 | Zbl 1248.16007
[16] Yang, X. Y., Chen, W. J.: 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] Yang, X. Y., Ding, N. Q.: On a question of Gillespie. Forum Math. 27 (2015), 3205-3231. DOI 10.1515/forum-2013-6014 | MR 3420339 | Zbl 1347.18003
[18] Yang, X. Y., Liu, Z. K.: Gorenstein projective, injective, and flat complexes. Commun. Algebra 39 (2011), 1705-1721. DOI 10.1080/00927871003741497 | MR 2821502 | Zbl 1238.16002
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