Geodesic
Gorenstein projective complexes with respect to cotorsion pairs
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 117-129 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/CMJ.2018.0194-17
Classification : 18G25, 18G35
Keywords: cotorsion pair; Gorenstein projective complex with respect to cotorsion pairs; stability of Gorenstein categories 
@article{10_21136_CMJ_2018_0194_17,
     author = {Zhao, Renyu and Ma, Pengju},
     title = {Gorenstein projective complexes with respect to cotorsion pairs},
     journal = {Czechoslovak Mathematical Journal},
     pages = {117--129},
     year = {2019},
     volume = {69},
     number = {1},
     doi = {10.21136/CMJ.2018.0194-17},
     mrnumber = {3923579},
     zbl = {07088774},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0194-17/}
}
TY  - JOUR
AU  - Zhao, Renyu
AU  - Ma, Pengju
TI  - Gorenstein projective complexes with respect to cotorsion pairs
JO  - Czechoslovak Mathematical Journal
PY  - 2019
SP  - 117
EP  - 129
VL  - 69
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0194-17/
DO  - 10.21136/CMJ.2018.0194-17
LA  - en
ID  - 10_21136_CMJ_2018_0194_17
ER  - 
%0 Journal Article
%A Zhao, Renyu
%A Ma, Pengju
%T Gorenstein projective complexes with respect to cotorsion pairs
%J Czechoslovak Mathematical Journal
%D 2019
%P 117-129
%V 69
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2018.0194-17/
%R 10.21136/CMJ.2018.0194-17
%G en
%F 10_21136_CMJ_2018_0194_17
Zhao, Renyu; Ma, Pengju. Gorenstein projective complexes with respect to cotorsion pairs. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 1, pp. 117-129. doi: 10.21136/CMJ.2018.0194-17

[1] Bouchiba, S.: Stability of Gorenstein classes of modules. Algebra Colloq. 20 (2013), 623-636. | DOI | MR | JFM

[2] Bouchiba, S., Khaloui, M.: Stability of Gorenstein flat modules. Glasg. Math. J. 54 (2012), 169-175. | DOI | MR | JFM

[3] Bravo, D., Gillespie, J.: Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44 (2016), 2213-2233. | DOI | MR | JFM

[4] Enochs, E. E., Rozas, J. R. García: Gorenstein injective and projective complexes. Commun. Algebra 26 (1998), 1657-1674. | DOI | MR | JFM

[5] Enochs, E. E., Jenda, O. M. G.: Gorenstein injective and projective modules. Math. Z. 220 (1995), 611-633. | DOI | MR | JFM

[6] Enochs, E. E., Jenda, O. M. G.: Relative Homological Algebra. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin (2000). | DOI | MR | JFM

[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 | JFM

[8] Gillespie, J.: The flat model structure on Ch($R$). Trans. Am. Math. Soc. 356 (2004), 3369-3390. | DOI | MR | JFM

[9] Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra 189 (2004), 167-193. | DOI | MR | JFM

[10] Hu, J. S., Xu, A. M.: On stability of F-Gorenstein flat categories. Algebra Colloq. 23 (2016), 251-262. | DOI | MR | JFM

[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 | MR | JFM

[12] Sather-Wagstaff, S., Sharif, T., White, D.: Stability of Gorenstein categories. J. Lond. Math. Soc., II. Ser. 77 (2008), 481-502. | DOI | MR | JFM

[13] Xu, A. M., Ding, N. Q.: On stability of Gorenstein categories. Commun. Algebra 42 (2013), 3793-3804. | DOI | MR | JFM

[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 | MR | JFM

[15] Yang, G., K.Liu, Z.: Stability of Gorenstein flat categories. Glasg. Math. J. 54 (2012), 177-191. | DOI | MR | JFM

[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 | MR | JFM

[17] Yang, X. Y., Ding, N. Q.: On a question of Gillespie. Forum Math. 27 (2015), 3205-3231. | DOI | MR | JFM

[18] Yang, X. Y., Liu, Z. K.: Gorenstein projective, injective, and flat complexes. Commun. Algebra 39 (2011), 1705-1721. | DOI | MR | JFM

Cité par Sources :