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:
the horizontal arrows are surjective,
if the characteristic of A/\mathfrak m_ A is zero, then S and R are power series rings over fields,
if the characteristic of A/\mathfrak m_ A is p > 0, then S and R are power series rings over Cohen rings, and
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
Comments (2)
Comment #6717 by Alex Scheffelin on
Comment #6792 by Alex Scheffelin on