FP\(_n\)-Injective and FP\(_n\)-Flat Complexes

  • Zenghui GaoEmail author
  • Jie Peng


Let R be an arbitrary ring and \(n\ge 0\) be an integer or \(n = \infty \). We introduce and study FP\(_n\)-injective and FP\(_n\)-flat complexes of modules, which unify the following notions: (FP-)injective and flat complexes (Yang and Liu in Commun Algebra 38:131–142, 2010; Enochs and García Rozas in J Algebra 210:86–102, 1998), absolutely clean and level complexes (Bravo and Gillespie in Commun Algebra 44:2213–2233, 2016) and weak injective and weak flat complexes (Gao and Huang in Glasgow Math J 58:539–557, 2016). Suppose that \(n>1\) is an integer. It is shown that a complex C is FP\(_n\)-injective (resp. FP\(_n\)-flat) if and only if C is exact and all cycles of C are FP\(_n\)-injective (resp. FP\(_n\)-flat) as R-modules. Then we investigate duality pairs relative to the FP\(_n\)-injective and FP\(_n\)-flat complexes. It is shown that the pairs (fp\(_n\mathcal {I}\), fp\(_n\mathcal {F}\)) and (fp\(_n\mathcal {F}\), fp\(_n\mathcal {I}\))) are duality pairs, where fp\(_n\mathcal {I}\) and fp\(_n\mathcal {F}\) denote the subcategories of FP\(_n\)-injective and FP\(_n\)-flat complexes, respectively. As applications, we get that any complex admits an FP\(_n\)-injective (resp. FP\(_n\)-flat) cover and preenvelope. In addition, cotorsion pairs associated with the FP\(_n\)-injective and FP\(_n\)-flat complexes are considered.


Finitely n-presented complexes FP\(_n\)-injective complexes FP\(_n\)-flat complexes Duality pairs Cotorsion pairs Covers Preenvelopes 

Mathematics Subject Classification

18G35 18G15 



This research was partially supported by National Natural Science Foundation of China (11571164 and 11671283), Science and Technology Foundation of Sichuan Province (2017JY0131) and Major Project of Education Department of Sichuan Province (17ZA0058). The authors thank the referee for the useful suggestions and thank Professor Xiaoyan Yang for sending us the paper [22].


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Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  1. 1.College of Applied MathematicsChengdu University of Information TechnologyChengduPeople’s Republic of China

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