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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