Remark 1

Let $A$ be a noetherian ring and $a$ an ideal of $A$. Suppose that $A$ is separated and complete for the $a$-adic topology, and consider the ring $R = A \{ X_1, ..., X_n \}$ of restricted formal power series (III, $§ 4$, No. 2, Definition 2). Then $\dim(R) = \dim(A) + n$. In fact, $R$ is the completion of the ring $B = A[X_1, ..., X_n]$ for the $aB$-adic topology, whence $\dim(R) \leqslant \dim(A[X_1, ..., X_n]) = \dim(A) + n$. The inverse inequality is proved as in the case of formal power series.