Relative FP-injective and FP-flat complexes and their model structures

Abstract

The present article studies homological and homotopical aspects of FP_n-injective and FP_n-flat complexes, and describes homological dimensions associated to them. After establishing a relation between these dimensions via Pontrjagin duality, we construct (pre)covers and (pre)envelopes by complexes with finite FP_n-injective and FP_n-flat dimensions, thus setting up the bases for an approximation theory relative to complexes of finite type. We also construct on the category of complexes several model structures from modules and complexes with finite FP_n-injective and FP_n-flat dimensions, and analyze some situations where it is possible to connect these structures via Quillen functors.

Keywords:

• Complex
• preenvelope
• cover
• cotorsion pair
• model structure
• Quillen equivalence

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 18G15
• 18G35
• 18G55
• 16E05
• 16E10
• 16E35

Acknowledgements

The authors thank Professor Zhaoyong Huang for his careful guidance and helpful suggestions. Special thanks to the referee whose valuable corrections and suggestions have greatly improved the presentation and organization of this article.

Additional information

Funding

The first author was supported by the National Natural Science Foundation of China (No. 11571164), Postgraduate Research and Innovation Program of Jiangsu Province 2016 (No. KYZZ16 _0034), and Nanjing University Innovation and Creative Program for PhD candidate (No. 2016011). The second author was supported by a DGAPA-UNAM (Dirección General de Asuntos del Personal Académico - Universidad Nacional Autónoma de México) postdoctoral fellowship.