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

In this paper, we introduce the notions of ${\mathrm{F}\mathrm{P}}_n$-injective and ${\rm FP}_n$-flatcomplexesintermsofcomplexesoftype${\rm FP}_n$. We show that some characterizations analogous to that of injective, FP-injective and flat complexes exist for ${\mathrm{F}\mathrm{P}}_n$-injective and ${\mathrm{F}\mathrm{P}}_n$-flat complexes. We also introduce and study ${\mathrm{F}\mathrm{P}}_n$-injective and ${\mathrm{F}\mathrm{P}}_n$-flat dimensions of modules and complexes, and give a relation between them in terms of Pontrjagin duality. The existence of pre-envelopes and covers in this setting is discussed, and we prove that any complex has an ${\mathrm{F}\mathrm{P}}_n$-flat cover and an ${\mathrm{F}\mathrm{P}}_n$-flat pre-envelope, and in the case $n\ge 2$ that any complex has an ${\mathrm{F}\mathrm{P}}_n$-injective cover and an ${\mathrm{F}\mathrm{P}}_n$-injective pre-envelope. Finally, we construct model structures on the category of complexes from the classes of modules with bounded ${\mathrm{F}\mathrm{P}}_n$-injective and ${\mathrm{F}\mathrm{P}}_n$-flat dimensions, and analyze several conditions under which it is possible to connect these model structures via Quillen functors and Quillen equivalences.