Example 1

Tag 0DP2, permanent. Cite it as [Bourbaki, Tag 0DP2].

Algebra, chapter X, § 1. Compléments d’algèbre linéaire

It is clear that $A_s$ is a flat $A$ -module; it follows then from prop. 4 (i) that every free $A$ -module, and more generally every projective $A$ -module, is flat (see also II, p. 63, cor. 6).

\* Conversely, every finitely presented flat $A$ -module is projective (No. 5). \*

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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