The Stacks project

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15.38 Some results on power series rings

Questions on formally smooth maps between Noetherian local rings can often be reduced to questions on maps between power series rings. In this section we prove some helper lemmas to facilitate this kind of argument.

Lemma 15.38.1. Let $K$ be a field of characteristic $0$ and $A = K[[x_1, \ldots , x_ n]]$. Let $L$ be a field of characteristic $p > 0$ and $B = L[[x_1, \ldots , x_ n]]$. Let $\Lambda $ be a Cohen ring. Let $C = \Lambda [[x_1, \ldots , x_ n]]$.

  1. $\mathbf{Q} \to A$ is formally smooth in the $\mathfrak m$-adic topology.

  2. $\mathbf{F}_ p \to B$ is formally smooth in the $\mathfrak m$-adic topology.

  3. $\mathbf{Z} \to C$ is formally smooth in the $\mathfrak m$-adic topology.

Proof. By the universal property of power series rings it suffices to prove:

  1. $\mathbf{Q} \to K$ is formally smooth.

  2. $\mathbf{F}_ p \to L$ is formally smooth.

  3. $\mathbf{Z} \to \Lambda $ is formally smooth in the $\mathfrak m$-adic topology.

The first two are Algebra, Proposition 10.152.9. The third follows from Algebra, Lemma 10.154.7 since for any test diagram as in Definition 15.36.1 some power of $p$ will be zero in $A/J$ and hence some power of $p$ will be zero in $A$. $\square$

Lemma 15.38.2. Let $K$ be a field and $A = K[[x_1, \ldots , x_ n]]$. Let $\Lambda $ be a Cohen ring and let $B = \Lambda [[x_1, \ldots , x_ n]]$.

  1. 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.

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

  3. 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.

  4. 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). $\square$

Lemma 15.38.3. Let $A \to B$ be a local homomorphism of Noetherian complete local rings. Then there exists a commutative diagram

\[ \xymatrix{ S \ar[r] & B \\ R \ar[u] \ar[r] & A \ar[u] } \]

with the following properties:

  1. the horizontal arrows are surjective,

  2. if the characteristic of $A/\mathfrak m_ A$ is zero, then $S$ and $R$ are power series rings over fields,

  3. if the characteristic of $A/\mathfrak m_ A$ is $p > 0$, then $S$ and $R$ are power series rings over Cohen rings, and

  4. $R \to S$ maps a regular system of parameters of $R$ to part of a regular system of parameters of $S$.

In particular $R \to S$ is flat (see Algebra, Lemma 10.127.2) with regular fibre $S/\mathfrak m_ R S$ (see Algebra, Lemma 10.105.3).

Proof. Use the Cohen structure theorem (Algebra, Theorem 10.154.8) to choose a surjection $S \to B$ as in the statement of the lemma where we choose $S$ to be a power series over a Cohen ring if the residue characteristic is $p > 0$ and a power series over a field else. Let $J \subset S$ be the kernel of $S \to B$. Next, choose a surjection $R = \Lambda [[x_1, \ldots , x_ n]] \to A$ where we choose $\Lambda $ to be a Cohen ring if the residue characteristic of $A$ is $p > 0$ and $\Lambda $ equal to the residue field of $A$ otherwise. We lift the composition $\Lambda [[x_1, \ldots , x_ n]] \to A \to B$ to a map $\varphi : R \to S$. This is possible because $\Lambda [[x_1, \ldots , x_ n]]$ is formally smooth over $\mathbf{Z}$ in the $\mathfrak m$-adic topology (see Lemma 15.38.1) by an application of Lemma 15.36.5. Finally, we replace $\varphi $ by the map $\varphi ' : R = \Lambda [[x_1, \ldots , x_ n]] \to S' = S[[y_1, \ldots , y_ n]]$ with $\varphi '|_\Lambda = \varphi |_\Lambda $ and $\varphi '(x_ i) = \varphi (x_ i) + y_ i$. We also replace $S \to B$ by the map $S' \to B$ which maps $y_ i$ to zero. After this replacement it is clear that a regular system of parameters of $R$ maps to part of a regular sequence in $S'$ and we win. $\square$

There should be an elementary proof of the following lemma.

Lemma 15.38.4. Let $S \to R$ and $S' \to R$ be surjective maps of complete Noetherian local rings. Then $S \times _ R S'$ is a complete Noetherian local ring.

Proof. Let $k$ be the residue field of $R$. If the characteristic of $k$ is $p > 0$, then we denote $\Lambda $ a Cohen ring (Algebra, Definition 10.154.5) with residue field $k$ (Algebra, Lemma 10.154.6). If the characteristic of $k$ is $0$ we set $\Lambda = k$. Choose a surjection $\Lambda [[x_1, \ldots , x_ n]] \to R$ (as in the Cohen structure theorem, see Algebra, Theorem 10.154.8) and lift this to maps $\Lambda [[x_1, \ldots , x_ n]] \to S$ and $\varphi : \Lambda [[x_1, \ldots , x_ n]] \to S$ and $\varphi ' : \Lambda [[x_1, \ldots , x_ n]] \to S'$ using Lemmas 15.38.1 and 15.36.5. Next, choose $f_1, \ldots , f_ m \in S$ generating the kernel of $S \to R$ and $f'_1, \ldots , f'_{m'} \in S'$ generating the kernel of $S' \to R$. Then the map

\[ \Lambda [[x_1, \ldots , x_ n, y_1, \ldots , y_ m, z_1, \ldots , z_{m'}]] \longrightarrow S \times _ R S, \]

which sends $x_ i$ to $(\varphi (x_ i), \varphi '(x_ i))$ and $y_ j$ to $(f_ j, 0)$ and $z_{j'}$ to $(0, f'_ j)$ is surjective. Thus $S \times _ R S'$ is a quotient of a complete local ring, whence complete. $\square$


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