The Stacks project

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. $\square$


Comments (2)

Comment #6717 by Alex Scheffelin on

Are we actually able to lift the map to a map in the situation where has characteristic ? It's claimed that as is formally smooth over in the -adic topology this is possible, but this would require to be continuous for the -adic topology, which I think in particular means that for some , but if has characteristic this will never be .

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.


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07NN. Beware of the difference between the letter 'O' and the digit '0'.