nLab Lazard's criterion

This page is about characterization of flat modules. For the characterization of the Lazard ring (in formal group laws) see instead at Lazard's theorem.

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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

The original article:

• Daniel Lazard, Sur les modules plats C. R. Acad. Sci. Paris 258 (1964) 6313-6316

Exposition:

• Robert Hines, Lazard’s theorem (characterizing flatness) (2016) [pdf, pdf]

• Stacks Project, Lazard’s theorem [tag:058G]