# Minimal complexes of cotorsion flat modules

## TL;DR

This paper establishes criteria for minimality of complexes of cotorsion flat modules over noetherian rings, utilizing Enochs' structure theory, and demonstrates that every module is isomorphic to a minimal semi-flat complex of such modules.

## Contribution

It provides new criteria for minimality of complexes of cotorsion flat modules and shows their applicability to representing modules as minimal semi-flat complexes.

## Key findings

- Criteria for minimal complexes of cotorsion flat modules
- Any complex built from covers or envelopes is minimal
- Every module is isomorphic to a minimal semi-flat complex of cotorsion flat modules

## Abstract

Let R be a commutative noetherian ring. We give criteria for a complex of cotorsion flat R-modules to be minimal, in the sense that every self homotopy equivalence is an isomorphism. To do this, we exploit Enochs' description of the structure of cotorsion flat R-modules. More generally, we show that any complex built from covers in every degree (or envelopes in every degree) is minimal, as well as give a partial converse to this in the context of cotorsion pairs. As an application, we show that every R-module is isomorphic in the derived category over R to a minimal semi-flat complex of cotorsion flat R-modules.