Cartan-Eilenberg Ding projective complexes

Year 2020, Volume: 49 Issue: 1, 362 - 370, 06.02.2020

Bo Lu Zhongkui Liu

https://doi.org/10.15672/hujms.552221

Abstract

In this article, Cartan-Eilenberg Ding projective complexes are introduced and investigated. It is shown that a complex $C$ is Cartan-Eilenberg Ding projective if and only if $C_n$ and $C_n/\mathrm{B}_n(C)$ are Ding projective in $R$-$\mathrm{Mod}$ for each $n\in\mathbb{Z}$ when $R$ is a Ding-Chen ring. Some applications are also given.

Keywords

Ding projective module, Ding projective complex, Cartan-Eilenberg Ding projective complex

References

• [1] H. Cartan and S. Eilenberg, Homological algebra, Princeton University Press, 1956.
• [2] N. Q. Ding and J. L. Chen, The flat dimensions of injective modules, Manuscripta Math. 78 (1), 165-177, 1993.
• [3] N. Q. Ding and J. L. Chen, Coherent rings with finite self-FP-injective dimension, Comm. Algebra 24 (9), 2963-2980, 1996.
• [4] N. Q. Ding, Y. L. Li and L. X. Mao, Strongly Gorenstein flat modules, J. Aust, Math. Soc. 86 (3), 323-338, 2009.
• [5] E. E. Enochs, Cartan-Eilenberg complexes and resolutions, J. Algebra 342 (1), 16-39, 2011.
• [6] J. Gillespie, The flat model structure on Ch(R), Trans. Amer. Math. Soc. 356 (8), 3369-3390, 2004.
• [7] J. Gillespie, Model structures on modules over Ding-Chen rings, Homology, Homotopy Appl. 12 (1), 61-73, 2010.
• [8] J. R. García Rozas, Covers and Envelopes in the Category of Complexes, Chapman & Hall/CRC, Boca Raton, FL, 1999.
• [9] B. Lu and Z. K. Liu, Cartan-Eilenberg FP-injective complexes, J. Aust, Math. Soc. 103 (3), 387-401, 2017.
• [10] B. Stenstr¨om, Coherent rings and FP-injective modules, J. London Math. Soc. 2 (2), 323-329, 1970.
• [11] J. L. Verdier, Des cat´egories d´eriv´ees des cat´egories ab´eliennes, Ast´erisque 239, 1997.
• [12] X. Y. Yang and Z. K. Liu, Gorenstein projective, injective, and flat complexes, Comm. Algebra 39 (5), 1705-1721, 2011.
• [13] G. Yang, Z. K. Liu and L. Liang, Model structures on categories of complexes over Ding-Chen rings, Comm. Algebra 41 (1), 50-69, 2013.
• [14] G. Yang, Z.K. Liu and L. Liang, Ding projective and Ding injective modules, Algebra Colloq. 20 (4), 601-602, 2013.