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

## TL;DR

This paper introduces and studies ${
m FP}_n$-injective and ${
m FP}_n$-flat complexes, establishing their properties, dimensions, and model structures, and explores their relationships via duality and Quillen functors.

## Contribution

It defines ${
m FP}_n$-injective and ${
m FP}_n$-flat complexes, characterizes their properties, and constructs related model structures on the category of complexes.

## Key findings

- Existence of ${
m FP}_n$-flat covers and pre-envelopes for all complexes.
- Existence of ${
m FP}_n$-injective covers and pre-envelopes for $n \\geq 2$.
- Construction of model structures from classes with bounded ${
m FP}_n$-injective and flat dimensions.

## 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.