Hostname: page-component-7dd5485656-2pp2p Total loading time: 0 Render date: 2025-10-31T12:22:19.880Z Has data issue: false hasContentIssue false
  • English
  • Français

FP-injective dimensions and Gorenstein homology

Part of: Homological algebra

Published online by Cambridge University Press:  21 December 2022

Gang Yang  and
Junpeng Wang
Gang Yang
Affiliation:
Department of Mathematics, Lanzhou Jiaotong University, Lanzhou 730070, China (yanggang@mail.lzjtu.cn)
Junpeng Wang*
Affiliation:
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China (wangjunpeng1218@163.com)
*
*Corresponding author
Get access Rights & Permissions [Opens in a new window]

Abstract

Let $R$ be a left coherent ring. It is proven that if an $R$-module $M$ has a finite FP-injective dimension, then the Gorenstein projective (resp. Gorenstein flat) dimension and the projective (resp. flat) dimension coincide. Also, we obtain that the pair ($\mathcal {GP},\, \mathcal {GP}^{\perp }$) forms a projective cotorsion pair under some mild conditions.

Keywords

FP-injective modulesGorenstein projective modulesGorenstein flat modules

MSC classification

Primary: 18G25: Relative homological algebra, projective classes
Secondary: 18G20: Homological dimension

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 65 , Issue 4 , November 2022 , pp. 1183 - 1199
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Auslander, M. and Bridger, M., Stable module theory, Mem. Amer. Math. Soc. Volume 94 (American Mathematical Society, Providence, RI, 1969).CrossRefGoogle Scholar
Bennis, D. and Mahdou, N., Global Gorenstein dimensions, Proc. Amer. Math. Soc. 138(2) (2010), 461–465.Google Scholar
Bican, L., El Bashir, R. and Enochs, E. E., All modules have flat covers, Bull. Lond. Math. Soc. 33 (2001), 385–390.CrossRefGoogle Scholar
Bouchiba, S., On Gorenstein flat dimension, J. Algebra Appl. 14 (2015), 1550096.CrossRefGoogle Scholar
Bravo, D., Gillespie, J. and Hovey, M., The stable module category of a general ring, arXiv:1405.5768Google Scholar
Christensen, L. W., Frankild, A. and Holm, H., On Gorenstein projective, injective and flat dimensions – a functorial description with applications, J.Algebra 302 (2006), 231–279.CrossRefGoogle Scholar
Christensen, L. W., Foxby, H-B. and Holm, H., Derived category methods in commutative algebra, preprint, 2022.Google Scholar
Ding, N. Q. and Chen, J. L., The flat dimension of injective modules, Manuscritpa Math. 78 (1993), 165–177.CrossRefGoogle Scholar
Enochs, E. E. and Jenda, O. M. G., Gorenstein injective and projective modules, Math. Z. 220 (1995), 611–633.CrossRefGoogle Scholar
Ding, N. Q., Li, Y. L. and Mao, L. X., Strongly Gorenstein flat modules, J. Aust. Math. Soc. 86 (2009), 323–338.CrossRefGoogle Scholar
Enochs, E. E. and Jenda, O. M. G., Relative homological algebra, In de Gruyter Expositions in Mathematics, Volume 30 (Walter de Gruyter and Co, Berlin, 2000).CrossRefGoogle Scholar
Enochs, E. E., Jenda, O. M. G. and Torrecillas, B., Gorenstein flat modules, Nanjing Daxue Xuebao Shuxue Bannian Kan 10 (1993), 1–9.Google Scholar
Enochs, E. E. and Lépez-Ramos, J. A., Gorenstein flat modules (Nova Science Publishers Inc., New York, 2001).Google Scholar
Gillespie, J., Model Structures on Modules over Ding-Chen rings, Homology Homology Appl. 12 (2010), 61–73.Google Scholar
Gillespie, J., Gorenstein complexes and recollements from cotorsion pairs, Adv. Math. 291 (2016), 859–911.CrossRefGoogle Scholar
Gillespie, J., On Ding injective, Ding projective and Ding flat modules and complexes, Rocky Mountain J. Math. 47(8) (2017), 2641–2673.Google Scholar
Holm, H., Gorenstein projective, injective and flat modules, MSc Thesis, Institute for Mathematical Sciences (University of Copenhagen 2000).Google Scholar
Holm, H., Rings with finite Gorenstein injective dimension, Proc. Amer. Math. Soc. 132 (2004), 1279–1283.CrossRefGoogle Scholar
Holm, H., Gorenstein homological dimensions, J. Pure Appl. Algebra 189 (2004), 167–193.CrossRefGoogle Scholar
Iacob, A., Projectively coresolved Gorenstein flat and Ding projective modules, Comm. Algebra 48 (2020), 2883–2893.CrossRefGoogle Scholar
Jørgensen, P., Existence of Gorenstein projective resolutions and Tate cohomology, J. Eur. Math. Soc. 9(1) (2007), 59–76.CrossRefGoogle Scholar
Kirkman, E. and Kuzmanovich, J., On the global dimension of fibre products, Pacific J. Math. 134 (1988), 121–132.CrossRefGoogle Scholar
Krause, H., The stable derived category of a Noetherian scheme, Compos. Math. 141 (2005), 1128–1162.Google Scholar
Mao, L. X. and Ding, N. Q., Envelopes and covers by modules of finite FP-injective and flat dimensions, Comm. Algebra 35 (2007), 833–849.CrossRefGoogle Scholar
Murfet, D. and Salarian, S., Totally acyclic complexes over Noetherian schemes, Adv. Math. 226(2) (2011), 1096–1133.CrossRefGoogle Scholar
Salce, L., Cotorsion theories for abelian groups. Symposia Math. Volume 23 (1979).Google Scholar
Šaroch, J. and Št'ovíček, J., Singular compactness and definability for $\Sigma$-cotorsion and Gorenstein modules, Sel. Math. 26 (2020), 23.Google Scholar
Small, L. W., Hereditary rings, Proc. Amer. Math. Soc. 1(55) (1966), 25–27.Google Scholar
Stenström, B., Coherent rings and FP-injective modules, J. Lond. Math. Soc. 2(2) (1970), 323–329.CrossRefGoogle Scholar
Wang, J. P., Ding projective dimension of Gorenstein flat modules, Bull. Korean Math. Soc. 54 (2017), 1935–1950.Google Scholar
Wang, J. and Liang, L., A characterization of Gorenstein projective modules, Comm. Algebra 44 (2016), 1420–1432.CrossRefGoogle Scholar
Yang, G. and Liang, L., All modules have Gorenstein flat precovers, Comm. Algebra 42 (2014), 3078–3085.CrossRefGoogle Scholar