Abstract
We introduce and study the notions of restricted projective, injective and flat complexes of modules. It is shown in the paper that a complex C of modules is restricted projective if and only if for each \(m\in \mathbb {Z}\) the module \(C^{m}\) is restricted projective. A similar result for restricted injective complexes is also given. As an application, we characterize the complex of strongly torsion-free modules introduced by Xu.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aldrich, S.T., Enochs, E.E., García Rozas, J.R., Oyonarte, L.: Covers and envelopes in Grothendieck categories: flat covers of complexes with applications. J. Algebra 243, 615–630 (2001)
Christensen, L.W., Foxby, H.-B., Frankild, A.: Restricted homological dimensions and Cohen–Macaulayness. J. Algebra 251, 479–502 (2002)
Emmanouil, I., Talelli, O.: On the flat length of injective moudles. J. Lond. Math. Soc. 84, 408–432 (2011)
Enochs, E.E., García Rozas, J.R.: Gorenstein injective and projective complexes. Comm. Algebra 26, 1657–1674 (1998)
Enochs, E.E., Iacob, A.: The \(\aleph _{1}\)-product of DG-injective complexes. Proc. Edinb. Math. Soc. 49, 257–266 (2006)
Enochs, E.E., Jenda, O.M.G.: Relative homological algebra, de Gruyter Expositions in Mathematics, vol. 30. Walter de Gruyter & Co., Berlin (2000)
Enochs, E.E., Jenda, O.M.G., Xu, J.: Orthogonality in the category of complexes. Math. J. Okayama Univ. 38, 25–46 (1996)
García Rozas, J.R.: Covers and envelopes in the category of complexes of modules. CRC Press, Boca Raton, London,New York,Washington, D. C. (1999)
Gillespie, J.: The flat model structure on \(Ch(R)\). Trans. Am. Math. Soc. 356, 3369–3390 (2004)
Holm, H.: Gorenstein derived functors. Proc. Am. Math. Soc. 132, 1913–1923 (2004)
Holm, H.: Gorenstein homological dimensions. J. Pure Appl. Algebra 189, 167–193 (2004)
Jensen, C.U.: On the vanishing of \(\underleftarrow{\lim }^{(i)}\). J. Algebra. 15, 151–166 (1970)
Liang, L., Wu, D.: On the restricted projective dimension of complexes. Ukrainian Math. J. 65, 1042–1053 (2013)
Osofsky, B.L.: Homological dimension and cardinality. Trans. Am. Math. Soc. 151, 641–649 (1970)
Sharif, T., Yassemi, S.: Depth formulas, restricted Tor-dimension under base change. Rocky Mountain J. Math 34, 1131–1146 (2004)
Wei, J.: Finitistic dimension and restricted flat dimension. J. Algebra 320, 116–127 (2008)
Wu, D., Liu, Z.: On the restricted injective dimension of complexes. Comm. Algebra 41, 462–470 (2013)
Wu, D.: Finitistic dimension and restricted injective dimension. Czechoslovak Math. J. 65, 1023–1031 (2015)
Xu, J.: Flat covers of modules. Lecture notes in math, vol. 1634. Springer, Berlin (1996)
Yang, X.Y., Liu, Z.K.: Gorenstein projective, injective and flat complexes. Comm. Algebra 39, 1705–1721 (2011)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Siamak Yassemi.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research was partly supported by the National Natural Science Foundation of China (Grant No. 11761045), the Foundation of A Hundred Youth Talents Training Program of Lanzhou Jiaotong University, and the Natural Science Foundation of Gansu Province (Grant No. 18JR3RA113).
Rights and permissions
About this article
Cite this article
Liang, L., Song, Y. Restricted Projective, Injective and Flat Complexes. Bull. Iran. Math. Soc. 47, 1363–1378 (2021). https://doi.org/10.1007/s41980-020-00446-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-020-00446-x
Keywords
- Restricted projective complex
- Restricted injective complex
- Restricted flat complex
- Strongly torsion-free module