Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T20:02:25.951Z Has data issue: false hasContentIssue false
  • English
  • Français

WEAK INJECTIVE AND WEAK FLAT COMPLEXES

Part of: Homological algebra

Published online by Cambridge University Press:  21 July 2015

ZENGHUI GAO  and
ZHAOYONG HUANG
ZENGHUI GAO
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. China College of Mathematics, Chengdu University of Information Technology, Chengdu 610225, Sichuan Province, P.R. China e-mail: gaozenghui@cuit.edu.cn
ZHAOYONG HUANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P.R. China e-mail: huangzy@nju.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

MSC classification

Primary: 18G35: Chain complexes
Secondary: 18G15: Ext and Tor, generalizations, Künneth formula 18G20: Homological dimension
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 58 , Issue 3 , September 2016 , pp. 539 - 557
Copyright
Copyright © Glasgow Mathematical Journal Trust 2015 

References

REFERENCES

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.CrossRefGoogle Scholar
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. Angeleri Hügel, L., Herbera, D. and Trlifaj, J., Tilting modules and Gorenstein rings, Forum Math. 18 (2006), 211–229.Google Scholar
4. Avramov, L. L. and Foxby, H. B., Homological dimensions of unbounded complexes, J. Pure Appl. Algebra 71(1991), 129–155.CrossRefGoogle Scholar
5. Brown, K. S., Cohomology of groups (Springer-Verlag, New York, 1982).Google Scholar
6. Crawley-Boevey, W., Locally finitely presented additive categories, Comm. Algebra 22 (1994), 1641–1674.CrossRefGoogle Scholar
7. Crivei, S., Prest, M. and Torrecillas, B., Covers in finitely accessible categories, Proc. Amer. Math. Soc. 138 (2010), 1213–1221.CrossRefGoogle Scholar
8. Enochs, E. E., Injective and flat covers, envelopes and resolvents, Israel J. Math. 39 (1981), 189–209.Google Scholar
9. Enochs, E. E. and García Rozas, J. R., Tensor products of complexes, Math. J. Okayama Univ. 39 (1997), 17–39.Google Scholar
10. Enochs, E. E. and García Rozas, J. R., Flat covers of complexes, J. Algebra 210 (1998), 86–102.Google Scholar
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. Gao, Z. H. and Huang, Z. Y., Weak injective covers and dimension of modules, Acta Math. Hungar. (to appear).Google Scholar
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. Gao, Z. H. and Wang, F. G., Weak injective and weak flat modules, Comm. Algebra 43 (2015), 3857–3868.Google Scholar
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. Gillespie, J., The flat model structure on Ch(R), Trans. Amer. Math. Soc. 356 (2004), 3369–3390.Google Scholar
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. Hummel, L. and Marley, T., The Auslander-Bridger formula and the Gorenstein property for coherent rings, J. Commut. Algebra 1 (2009), 283–314.CrossRefGoogle Scholar
19. Rotman, J. J., An Introduction to Homological Algebra (Academic Press, New York, 1979).Google Scholar
20. Stenström, B., Rings of quotients (Springer-Verlag, New York, 1975).CrossRefGoogle Scholar
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. Weibel, C., An introduction to homological algebra, CSAM, vol. 38 (Cambridge University Press, Cambridge, 1994).Google Scholar
23. Yang, X. Y. and Liu, Z. K., FP-injective complexes, Comm. Algebra 38 (2010), 131–142.CrossRefGoogle ScholarPubMed