The Stacks project

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. $\square$


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