# Countably generated flat modules are quite flat

## TL;DR

This paper proves that over certain commutative Noetherian rings, all countably generated flat modules are quite flat, and extends this to broader classes of rings, providing new insights into the structure of flat modules.

## Contribution

It establishes that countably generated flat modules are quite flat over Noetherian rings and introduces the CFQ property for rings, characterizing various classes of rings in terms of flat modules.

## Key findings

- Countably generated flat modules are quite flat over Noetherian rings.
- All von Neumann regular and S-almost perfect rings are CFQ.
- Characterizations of CFQ rings include perfectness for zero-dimensional local rings and strong discreteness for valuation domains.

## Abstract

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of $R$ in countable multiplicative subsets. We also show that if the spectrum of $R$ is of cardinality less than $\kappa$, where $\kappa$ is an uncountable regular cardinal, then every flat $R$-module is a transfinite extension of flat modules with less than $\kappa$ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. More generally, we say that a commutative ring is CFQ if every countably presented flat $R$-module is quite flat. We show that all von Neumann regular rings and all $S$-almost perfect rings are CFQ. A zero-dimensional local ring is CFQ if and only if it is perfect. A domain is CFQ if and only if all its proper quotient rings are CFQ. A valuation domain is CFQ if and only if it is strongly discrete.