The direct limit closure of perfect complexes

TL;DR

This paper extends classical module theory results to complexes, showing that certain complexes can be expressed as direct limits, and applies these findings to reprove existing theorems in the field.

Contribution

It proves an analogous direct limit result for complexes of modules, expanding the classical theory from modules to complexes.

Findings

Complexes of modules can be expressed as direct limits of finitely generated free complexes.

Reproves key results by Enochs and García Rozas, and Neeman using the new framework.

Establishes foundational properties of complexes related to direct limits.

Abstract

Every projective module is flat. Conversely, every flat module is a direct limit of finitely generated free modules; this was proved independently by Govorov and Lazard in the 1960s. In this paper we prove an analogous result for complexes of modules, and as applications we reprove some results due to Enochs and Garc\'ia Rozas and to Neeman.