Corollary 3

If $A$ is a commutative Noetherian ring, the ring of formal power series $A[[X_1, \ldots, X_n]]$ is a flat $A$-module.

It is the completion of the polynomial ring

with respect to the $m$-adic topology, where $m$ is the set of polynomials with no constant term ($§ 2,$ no. 12, Example 1); as $B$ is Noetherian ($§ 2,$ no. 10, Corollary 2 to Theorem 2), $A[[X_1, \ldots, X_n]]$ is a flat $B$-module by Theorem 3 and, as $B$ is free $A$-module, $A[[X_1, \ldots, X_n]]$ is a flat $A$-module (Chapter I, § 2, no. 7, Corollary 3 to Proposition 8).