# nLab Lazard's criterion

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Homological algebra

homological algebra

and

nonabelian homological algebra

diagram chasing

# Contents

## Statement

Let $R$ be a commutative ring. (or maybe any ring?)

###### Proposition

(Lazard’s criterion)

An $R$-module is a flat module precisely if it is a filtered colimit of free modules.

This is due to (Lazard (1964)). See at flat module for more.

## References

• Daniel Lazard, Sur les modules plats C. R. Acad. Sci. Paris 258, 6313–6316 (1964)
Revised on November 3, 2015 06:39:18 by Anonymous Coward (129.67.186.166)