Corollary

With the hypotheses and notation of Proposition 5, suppose that $s \geq 1$, so that $f \in Bm + BX$. Then the A-homomorphism $h$ of $B' = A[[T]]$ to $B = A[[X]]$ such that $h(T) = f$ (Chapter III, § 2, no. 9, Proposition 11 (a)) defines on B a free $B'$-module structure admitting $\{1, X, \ldots, X^{s-1}\}$ as basis. In particular $h$ is injective.

Let the $B'$-module B be given the (T)-adic filtration, which consists of the $f^n B$ for $n \geq 0$ (Chapter 111, § 2, no. 1). Then $B/fB$ is a free module over the ring $A = B'/TB'$ and the images of the $X^i$ ($0 \leq i \leq s-1$) in this A-module form a basis of it (Proposition 5); as moreover $f$ is not a divisor of zero in B (Proposition 5), $Bf^n/Bf^{n+1}$ is also a free $(B'/TB')$-module of rank s, so that condition (GR) of Chapter 111, § 2, no. 8 is satisfied (replacing A by B' and M by B). On the other hand, since B' is Hausdorff and complete with respect to the (T)-adic filtration and gr(B) is a finitely generated gr(B')-module by the above, it is seen first (Chapter III, § 2, no. 9, Corollary 1 to Proposition 12) that B is a finitely generated B' module. The first assertion of the corollary then follows from Chapter III, § 2, no. 9, Proposition 13. The second follows immediately from it.