Title: Relative FP-injective and FP-flat complexes and their model structures,
Abstract: In this paper, we introduce the notions of ${\rm FP}_n$-injective and ${\rm
FP}_n$-flat complexes in terms of complexes of type ${\rm FP}_n$. We show that
some characterizations analogous to that of injective, FP-injective and flat
complexes exist for ${\rm FP}_n$-injective and ${\rm FP}_n$-flat complexes. We
also introduce and study ${\rm FP}_n$-injective and ${\rm FP}_n$-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 ${\rm FP}_n$-flat
cover and an ${\rm FP}_n$-flat pre-envelope, and in the case $n \geq 2$ that
any complex has an ${\rm FP}_n$-injective cover and an ${\rm FP}_n$-injective
pre-envelope. Finally, we construct model structures on the category of
complexes from the classes of modules with bounded ${\rm FP}_n$-injective and
${\rm FP}_n$-flat dimensions, and analyze several conditions under which it is
possible to connect these model structures via Quillen functors and Quillen
equivalences.