The Stacks project

Lemma 15.37.9. Let $R$, $S$ be rings. Let $\mathfrak n \subset S$ be an ideal. Let $R \to R'$ be a ring map. Set $S' = S \otimes _ R R'$ and $\mathfrak n' = \mathfrak nS$. If

  1. the map $R \to R'$ embeds $R$ as a direct summand of $R'$ as an $R$-module, and

  2. $R' \to S'$ is formally smooth for the $\mathfrak n'$-adic topology,

then $R \to S$ is formally smooth in the $\mathfrak n$-adic topology.

Proof. Let a solid diagram

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

as in Definition 15.37.1 be given. Set $A' = A \otimes _ R R'$ and $J' = \mathop{\mathrm{Im}}(J \otimes _ R R' \to A')$. The base change of the diagram above is the diagram

\[ \xymatrix{ S' \ar[r] \ar@{-->}[rd]^{\psi '} & A'/J' \\ R' \ar[u] \ar[r] & A' \ar[u] } \]

with continuous arrows. By condition (2) we obtain the dotted arrow $\psi ' : S' \to A'$. Using condition (1) choose a direct summand decomposition $R' = R \oplus C$ as $R$-modules. (Warning: $C$ isn't an ideal in $R'$.) Then $A' = A \oplus A \otimes _ R C$. Set

\[ J'' = \mathop{\mathrm{Im}}(J \otimes _ R C \to A \otimes _ R C) \subset J' \subset A'. \]

Then $J' = J \oplus J''$ as $A$-modules. The image of the composition $\psi : S \to A'$ of $\psi '$ with $S \to S'$ is contained in $A + J' = A \oplus J''$. However, in the ring $A + J' = A \oplus J''$ the $A$-submodule $J''$ is an ideal! (Use that $J^2 = 0$.) Hence the composition $S \to A + J' \to (A + J')/J'' = A$ is the arrow we were looking for. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 15.37: Formally smooth maps of topological rings

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Your email address will not be published.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 07EH. Beware of the difference between the letter 'O' and the digit '0'.