Lemma 15.39.2. Let $K$ be a field and $A = K[[x_1, \ldots , x_ n]]$. Let $\Lambda $ be a Cohen ring and let $B = \Lambda [[x_1, \ldots , x_ n]]$.

If $y_1, \ldots , y_ n \in A$ is a regular system of parameters then $K[[y_1, \ldots , y_ n]] \to A$ is an isomorphism.

If $z_1, \ldots , z_ r \in A$ form part of a regular system of parameters for $A$, then $r \leq n$ and $A/(z_1, \ldots , z_ r) \cong K[[y_1, \ldots , y_{n - r}]]$.

If $p, y_1, \ldots , y_ n \in B$ is a regular system of parameters then $\Lambda [[y_1, \ldots , y_ n]] \to B$ is an isomorphism.

If $p, z_1, \ldots , z_ r \in B$ form part of a regular system of parameters for $B$, then $r \leq n$ and $B/(z_1, \ldots , z_ r) \cong \Lambda [[y_1, \ldots , y_{n - r}]]$.

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