Lemma 15.39.3. 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 10.128.2) with regular fibre $S/\mathfrak m_ R S$ (see Algebra, Lemma 10.106.3).

Proof. Use the Cohen structure theorem (Algebra, Theorem 10.160.8) 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 15.39.1) by an application of Lemma 15.37.5. 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. $\square$

Comment #6717 by Alex Scheffelin on

Are we actually able to lift the map $f\colon\Lambda[[x_1,\dots,x_n]]\to A\to B$ to a map $\varphi\colon R\to S$ in the situation where $B$ has characteristic $0$? It's claimed that as $\Lambda[[x_1,\dots,x_n]]$ is formally smooth over $\mathbb{Z}$ in the $\mathfrak{m}$-adic topology this is possible, but this would require $f$ to be continuous for the $\mathfrak{m}$-adic topology, which I think in particular means that $f(p^n) = 0$ for some $n$, but if $B$ has characteristic $0$ this will never be $0$.

Comment #6792 by Alex Scheffelin on

I realized what was wrong with my previous comment, I was confused about which topologies the map must be continuous for in the statement of Lemma 07NJ.

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