Geodesic
GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES
Glasgow mathematical journal, Tome 59 (2017) no. 3, pp. 685-703

Voir la notice de l'article provenant de la source Cambridge University Press

Given a complete hereditary cotorsion pair $(\mathcal{X}, \mathcal{Y})$ , we introduce the concept of $(\mathcal{X}, \mathcal{X} \cap \mathcal{Y})$ -Gorenstein projective modules and study its stability properties. As applications, we first get two model structures related to Gorenstein flat modules over a right coherent ring. Secondly, for any non-negative integer n, we construct a cofibrantly generated model structure on Mod(R) in which the class of fibrant objects are the modules of Gorenstein injective dimension ≤ n over a left Noetherian ring R. Similarly, if R is a left coherent ring in which all flat left R-modules have finite projective dimension, then there is a cofibrantly generated model structure on Mod(R) such that the cofibrant objects are the modules of Gorenstein projective dimension ≤ n. These structures have their analogous in the category of chain complexes. 
XU, AIMIN. GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES. Glasgow mathematical journal, Tome 59 (2017) no. 3, pp. 685-703. doi: 10.1017/S0017089516000483
@article{10_1017_S0017089516000483,
     author = {XU, AIMIN},
     title = {GORENSTEIN {MODULES} {AND} {GORENSTEIN} {MODEL} {STRUCTURES}},
     journal = {Glasgow mathematical journal},
     pages = {685--703},
     year = {2017},
     volume = {59},
     number = {3},
     doi = {10.1017/S0017089516000483},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000483/}
}
TY  - JOUR
AU  - XU, AIMIN
TI  - GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES
JO  - Glasgow mathematical journal
PY  - 2017
SP  - 685
EP  - 703
VL  - 59
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000483/
DO  - 10.1017/S0017089516000483
ID  - 10_1017_S0017089516000483
ER  - 
%0 Journal Article
%A XU, AIMIN
%T GORENSTEIN MODULES AND GORENSTEIN MODEL STRUCTURES
%J Glasgow mathematical journal
%D 2017
%P 685-703
%V 59
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089516000483/
%R 10.1017/S0017089516000483
%F 10_1017_S0017089516000483

[1] 1. Asadollahi, J. and Salarian, Sh., Cohomology theories based on flats, J. Algebra 353 (2012), 93–120. Google Scholar

[2] 2. Beligiannis, A. and Reiten, I., Homological and homotopical aspects of torsion theories, Mem. Amer. Math. Soc. 188, 2007. Google Scholar

[3] 3. Bennis, D., Weak Gorenstein global dimension, Int. Electron. J. Algebra. 8 (2010), 140–152. Google Scholar

[4] 4. Bennis, D. and Mahdou, N., Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138 (2) (2010), 461–465. Google Scholar

[5] 5. Bravo, D. and Gillespie, J., Absolutely clean, level, and Gorenstein AC-injective complexes, Comm. Algebra 44 (5) (2016), 2213–2233. Google Scholar

[6] 6. Bravo, D., Gillespie, J. and Hovey, M., The stable module category of a general ring, arXiv:1405.5768v1, 2014. Google Scholar

[7] 7. Christensen, L. W., Frankild, A. and Holm, H., On Gorenstein projective, injective and flat dimensions-A functorial description with applications, J. Algebra 302 (2006), 231–279. Google Scholar | DOI

[8] 8. Cortés Izurdiaga, M. and Guil Asensio, P. A., A Quillen model structure approach to the finitistic dimension conjectures, Math. Scand. 285 (7) (2012), 821–833. Google Scholar

[9] 9. Ding, N. and Chen, J., Coherent rings with finite self-FP-injective dimension, Comm. Algebra 24 (1996), 2963–2980. Google Scholar | DOI

[10] 10. Ding, N. and Chen, J., The flat dimensions of injective modules, Manuscripta Math. 78 (2) (1993), 165–177. Google Scholar

[11] 11. Enochs, E. E., Estrada, S. and García Rozas, J. R., Gorenstein categories and Tate cohomology on projective schemes, Math. Nachr. 281 (4) (2008), 525–540. Google Scholar

[12] 12. Enochs, E. E. and Jenda, O. M. G., Relative homological algebra (Walter de Gruyter, Berlin, New York, 2000). Google Scholar

[13] 13. Enochs, E. E., Jenda, O. M. G. and Lopez-Ramos, J. A., The existence of Gorenstein flat covers, Math. Scand. 94 (1) (2004), 46–62. Google Scholar | DOI

[14] 14. García Rozas, J. R., Covers and envelopes in the category of complexes of modules (BocaRaton, London, 1999). Google Scholar

[15] 15. Gillespie, J., How to construct a Hovey triple from two cotorsion pairs, preprint; arXiv:1406.2619, 2014. Google Scholar

[16] 16. Gillespie, J., Model structures on modules over Ding-Chen rings, Homology, Homotopy Appl. 12 (1) (2010), 61–73. Google Scholar

[17] 17. Gillespie, J., The flat model structure on Ch(R), Tran. Amer. Math. Soc. 356 (8) (2004), 3369–3390. Google Scholar | DOI

[18] 18. Gillespie, J., The flat stable module category of a coherent ring, arXiv:1412.4085v1, 2014. Google Scholar

[19] 19. Gillespie, J. and Hovey, M., Gorenstein model structures and generalized derived categories, Proc. Edinburgh Math. Soc. 53 (2010), 675–696. Google Scholar

[20] 20. Göbel, R. and Trlifaj, J., Approximations and endomorphism algebras of modules, GEM 41 (Walter de Gruyter, Berlin, New York, 2006). Google Scholar | DOI

[21] 21. Hashimoto, M., Auslander-Buchweitz Approximations of Equivariant Modules, London Mathematical Society Lecture Note Series, vol. 282 (Cambridge University Press, Cambridge, 2000). Google Scholar

[22] 22. Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (1–3) (2004), 167–193. Google Scholar

[23] 23. Hovey, M., Cotorsion pairs, model category structures, and representation theory, Math. Z. 241 (2002), 553–592. Google Scholar

[24] 24. Hovey, M., Cotorsion pairs and model categories, in Interactions between homotopy theory and algebra, Contemporary Mathematics, vol. 436 (Angeleri Hügel, L., Happel, D. and Krause, H., Editors) (American Mathematics Society, Providences, RI, 2007), 277–296. Google Scholar

[25] 25. Marco, A. and Pérez, B., Gorenstein homological dimensions and Abelian model structures, arxiv:1212.1517v3, 2014. Google Scholar

[26] 26. Pan, Q. X. and Cai, F. Q., -Gorenstein projective and injective modules, Turk. J. Math. 39 (2015), 81–90. Google Scholar

[27] 27. Rotman, J. J., An introduction to homological algebra (Academic Press, New York, 1979). Google Scholar

[28] 28. Salce, L., Cotorsion theories for abelian groups, Symposia Math. 23 (1979), 11–32. Google Scholar

[29] 29. Sather-Wagstaff, S., Sharif, T. and White, D., Stability of Gorenstein categories, J. London Math. Soc. 77 (2008), 481–502. Google Scholar

[30] 30. Sather-Wagstaff, S., Sharif, T. and White, D., AB-contexts and stability for Gorenstein flat modules with respect to semidualizing modules, Algebr. Represent. Theory 14 (2011), 403–428. Google Scholar | DOI

[31] 31. Trlifaj, J., Infinite dimensional tilting modules and cotorsion pairs, in Handbook of tilting theory, London Math. Soc. Lecture Note Ser., vol. 332 (Avramov, L. L., Christensen, J. D., Dwyer, W. G., Mandell, M. A. and Shipley, B. E., Editors) (Cambridge University Press, Cambridge, UK, 2007) 279–321. Google Scholar

[32] 32. Yang, C. H. and Liang, L., Gorenstein injective and projective complexes with respect to a semidualizing module, Comm. Algebra 40 (2012), 3352–3364. Google Scholar

[33] 33. Yang, X. Y. and Liu, Z. K., Gorenstein projective, injective, and flat complexes, Comm. Algebra 39 (2011), 1705–1721. Google Scholar

[34] 34. Yang, G. and Liu, Z. K., Stability of Gorenstein flat categories, Glasgow Math. J. 54 (1) (2012), 177–191. Google Scholar

[35] 35. Xu, J. Z., Flat covers of modules, Lecture Notes in Mathematics, vol. 1634 (Springer-Verlag, Berlin, 1996). Google Scholar

[36] 36. Zhu, X. S., The homological theory of quasi-resolving subcategories, J. Algebra 414 (2014), 6–40. Google Scholar

Cité par Sources :