Corollary

For an A-module M to be flat (resp. faithfully flat), it is necessary and sufficient that, for every maximal ideal m of A, M, be a flat (resp. faithfully flat) Am-module.*

The necessity of the conditions follows from Proposition 13. Conversely, if $M,$ is a flat $A_m$-module for every maximal ideal $m$ of $A$, $M$ is a flat A-module by virtue of Proposition 15 applied to the case where $\phi$ is the identity. Finally, if $M,$ is a faithfully flat $A_m$-module for all $m$, then $mM_m = mA_mM_m \neq M,$ hence $mM \neq M$ for all $m$ (no. 3, Proposition 9), which proves that $M$ is a faithfully flat A-module (Chapter I, § 3, no. 1, Proposition 1 (d)).