Geodesic
WEAK INJECTIVE AND WEAK FLAT COMPLEXES
Glasgow mathematical journal, Tome 58 (2016) no. 3, pp. 539-557

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

Let R be an arbitrary ring. We introduce and study a generalization of injective and flat complexes of modules, called weak injective and weak flat complexes of modules respectively. We show that a complex C is weak injective (resp. weak flat) if and only if C is exact and all cycles of C are weak injective (resp. weak flat) as R-modules. In addition, we discuss the weak injective and weak flat dimensions of complexes of modules. Finally, we show that the category of weak injective (resp. weak flat) complexes is closed under pure subcomplexes, pure epimorphic images and direct limits. As a result, we then determine the existence of weak injective (resp. weak flat) covers and preenvelopes of complexes. 
GAO, ZENGHUI; HUANG, ZHAOYONG. WEAK INJECTIVE AND WEAK FLAT COMPLEXES. Glasgow mathematical journal, Tome 58 (2016) no. 3, pp. 539-557. doi: 10.1017/S0017089515000324
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[1] 1. Aldrich, S. T., Enochs, E. E., García Rozas, J. R. and Oyonarte, L., Covers and envelopes in Grothendieck categories: Flat covers of complexes with applications, J. Algebra 243 (2001), 615–630. Google Scholar | DOI

[2] 2. Anderson, F. W. and Fuller, K. R., Rings and Categories of Modules, 2nd edition, Graduate Texts in Mathematics, vol. 13 (Springer-Verlag, New York, 1992). Google Scholar

[3] 3. Angeleri Hügel, L., Herbera, D. and Trlifaj, J., Tilting modules and Gorenstein rings, Forum Math. 18 (2006), 211–229. Google Scholar

[4] 4. Avramov, L. L. and Foxby, H. B., Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71(1991), 129–155. Google Scholar | DOI

[5] 5. Brown, K. S., Cohomology of groups (Springer-Verlag, New York, 1982). Google Scholar

[6] 6. Crawley-Boevey, W., Locally finitely presented additive categories, Comm. Algebra 22 (1994), 1641–1674. Google Scholar | DOI

[7] 7. Crivei, S., Prest, M. and Torrecillas, B., Covers in finitely accessible categories, Proc. Amer. Math. Soc. 138 (2010), 1213–1221. Google Scholar | DOI

[8] 8. Enochs, E. E., Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189–209. Google Scholar

[9] 9. Enochs, E. E. and García Rozas, J. R., Tensor products of complexes, Math. J. Okayama Univ. 39 (1997), 17–39. Google Scholar

[10] 10. Enochs, E. E. and García Rozas, J. R., Flat covers of complexes, J. Algebra 210 (1998), 86–102. Google Scholar

[11] 11. Enochs, E. E. and Jenda, O. M. G., Relative homological algebra, vol. 2, de Gruyter Expositions in Mathematics, vol. 54 (Walter de Gruyter GmbH & Co. KG, Berlin, 2011). Google Scholar

[12] 12. Gao, Z. H. and Huang, Z. Y., Weak injective covers and dimension of modules, Acta Math. Hungar. (to appear). Google Scholar

[13] 13. Gao, Z. H. and Wang, F. G., All Gorenstein hereditary rings are coherent, J. Algebra Appl. 13 (2014), 5 pages (1350140). Google Scholar

[14] 14. Gao, Z. H. and Wang, F. G., Weak injective and weak flat modules, Comm. Algebra 43 (2015), 3857–3868. Google Scholar

[15] 15. García Rozas, J. R., Covers and envelopes in the category of complexes of modules (Chapman & Hall/CRC, Boca Raton, London, 1999). Google Scholar

[16] 16. Gillespie, J., The flat model structure on Ch(R), Trans. Amer. Math. Soc. 356 (2004), 3369–3390. Google Scholar

[17] 17. Göbel, R. and Trlifaj, J., Approximations and endomorphism algebras of modules, de Gruyter Expositions in Mathematics, vol. 41, 2nd revised and extended edition (Walter de Gruyter GmbH & Co. KG, Berlin–Boston, 2012). Google Scholar

[18] 18. Hummel, L. and Marley, T., The Auslander-Bridger formula and the Gorenstein property for coherent rings, J. Commut. Algebra 1 (2009), 283–314. Google Scholar | DOI

[19] 19. Rotman, J. J., An Introduction to Homological Algebra (Academic Press, New York, 1979). Google Scholar

[20] 20. Stenström, B., Rings of quotients (Springer-Verlag, New York, 1975). Google Scholar | DOI

[21] 21. Wang, Z. P. and Liu, Z. K., FP-injective complexes and FP-injective dimension of complexes, J. Aust. Math. Soc. 91 (2011), 163–187. Google Scholar

[22] 22. Weibel, C., An introduction to homological algebra, CSAM, vol. 38 (Cambridge University Press, Cambridge, 1994). Google Scholar

[23] 23. Yang, X. Y. and Liu, Z. K., FP-injective complexes, Comm. Algebra 38 (2010), 131–142. Google Scholar PubMed | DOI

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