Remark 2

Let $A$ be a local noetherian ring, identified with a subring of its completion $\hat{A}$, and $B$ a subring of $\hat{A}$ containing $A$. Suppose that $B$ is local noetherian and that one has $m_A B = m_B$. Then the canonical injection of $A$ into $B$ extends to an isomorphism of $\hat{A}$ onto the completion $\hat{B}$ of $B$ (III, $§ 3$, No. 5, Proposition 11), whence $\dim(B) = \dim(A)$ (Corollary 2 to Proposition 8). This applies in particular to the following situation. \* Let $k$ be a complete non-discrete valued field, $A$ the local ring of the polynomial ring $k[X_1, ..., X_n]$ at the prime ideal generated by $X_1, ..., X_n$, and $B$ the ring of convergent power series in $X_1, ..., X_n$ with coefficients in $k$. Then the preceding hypotheses are satisfied, and consequently one has $\dim(B) = n$.*