Cartan-Eilenberg projective, injective and flat complexes. (English) Zbl 1374.18024

Summary: Let \(R\) be an associative ring with identity and \(\mathcal{F}\) a class of \(R\)-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg \(\mathcal{F}\) complexes and extend the basic properties of the class \(\mathcal{F}\) to the class \(\text{CE}(\mathcal{F}\)). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, \(\text{QF}\) rings, semisimple rings, hereditary rings and perfect rings.


18G10 Resolutions; derived functors (category-theoretic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18G35 Chain complexes (category-theoretic aspects), dg categories
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[1] F. W. Anderson, K. R. Fuller: Rings and Categories of Modules. Graduate Texts in Mathematics 13, Springer, New York, 1992. · Zbl 0765.16001
[2] H. Bass: Finitistic dimension and a homological generalization of semi-primary rings. Trans. Am. Math. Soc. 95 (1960), 466–488. · Zbl 0094.02201
[3] D. Bennis, N. Mahdou: Global Gorenstein dimensions. Proc. Amer. Math. Soc. 138 (2010), 461–465. · Zbl 1205.16007
[4] H. Cartan, S. Eilenberg: Homological Algebra. Princeton Mathematical Series 19, Princeton University Press 15, Princeton, 1999. · Zbl 0933.18001
[5] S. U. Chase: Direct products of modules. Trans. Amer. Math. Soc. 97 (1960), 457–473.
[6] T. J. Cheatham, D. R. Stone: Flat and projective character modules. Proc. Am. Math. Soc. 81 (1981), 175–177. · Zbl 0458.16014
[7] E. E. Enochs: Cartan-Eilenberg complexes and resolutions. J. Algebra 342 (2011), 16–39. · Zbl 1246.18005
[8] E. E. Enochs: Injective and flat covers, envelopes and resolvents. Isr. J. Math. 39 (1981), 189–209. · Zbl 0464.16019
[9] E. E. Enochs, O. M. G. Jenda: Relative Homological Algebra. Volume 1. De Gruyter Expositions in Mathematics 30, Walter de Gruyter, Berlin, 2011. · Zbl 1238.13002
[10] E. E. Enochs, J. R. García Rozas: Tensor products of complexes. Math. J. Okayama Univ. 39 (1997), 17–39.
[11] E. E. Enochs, J. A. López-Ramos: Kaplansky classes. Rend. Sem. Mat. Univ. Padova 107 (2002), 67–79.
[12] D. J. Fieldhouse: Character modules. Comment. Math. Helv. 46 (1971), 274–276. · Zbl 0219.16017
[13] J. R. García Rozas: Covers and Envelopes in the Category of Complexes of Modules. Chapman & Hall/CRC Research Notes in Mathematics 407, Chapman and Hall/CRC, Boca Raton, 1999.
[14] A. Iacob: DG-injective covers, #-injective covers. Commun. Algebra 39 (2011), 1673–1685. · Zbl 1222.18009
[15] J. J. Rotman: An Introduction to Homological Algebra. Universitext, Springer, Berlin, 2009.
[16] J. -L. Verdier: Derived Categories of Abelian Categories. Astérisque 239. Société Mathématique de France, Paris, 1996. (In French.)
[17] Z. Wang, Z. Liu: Complete cotorsion pairs in the category of complexes. Turk. J. Math. 37 (2013), 852–862. · Zbl 1285.18016
[18] J. Xu: Flat Covers of Modules. Lecture Notes in Mathematics 1634, Springer, Berlin, 1996. · Zbl 0860.16002
[19] G. Yang, L. Liang: Cartan-Eilenberg Gorenstein flat complexes. Math. Scand. 114 (2014), 5–25. · Zbl 1299.13018
[20] G. Yang, L. Liang: Cartan-Eilenberg Gorenstein projective complexes. J. Algebra Appl. 13 (2014), Article ID 1350068, 17 pages. · Zbl 1292.16005
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