Lemma 15.39.2 (07NM)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 15: More on Algebra
Section 15.39: Some results on power series rings
Lemma 15.39.2 (cite)

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 $B/(z_1, \ldots , z_ r) \cong \Lambda [[y_1, \ldots , y_{n - r}]]$ .

Proof. Proof of (1). Set $A' = K[[y_1, \ldots , y_ n]]$ . It is clear that the map $A' \to A$ induces an isomorphism $A'/\mathfrak m_{A'}^ n \to A/\mathfrak m_ A^ n$ for all $n \geq 1$ . Since $A$ and $A'$ are both complete we deduce that $A' \to A$ is an isomorphism. Proof of (2). Extend $z_1, \ldots , z_ r$ to a regular system of parameters $z_1, \ldots , z_ r, y_1, \ldots , y_{n - r}$ of $A$ . Consider the map $A' = K[[z_1, \ldots , z_ r, y_1, \ldots , y_{n - r}]] \to A$ . This is an isomorphism by (1). Hence (2) follows as it is clear that $A'/(z_1, \ldots , z_ r) \cong K[[y_1, \ldots , y_{n - r}]]$ . The proofs of (3) and (4) are exactly the same as the proofs of (1) and (2). $\square$

The Stacks project

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