Lemma 10.136.17. Let $R \to S$ be a smooth ring map. Given a commutative solid diagram

where $I \subset A$ is a locally nilpotent ideal, a dotted arrow exists which makes the diagram commute.

Lemma 10.136.17. Let $R \to S$ be a smooth ring map. Given a commutative solid diagram

\[ \xymatrix{ S \ar[r] \ar@{-->}[rd] & A/I \\ R \ar[r] \ar[u] & A \ar[u] } \]

where $I \subset A$ is a locally nilpotent ideal, a dotted arrow exists which makes the diagram commute.

**Proof.**
By Lemma 10.136.14 we can extend the diagram to a commutative diagram

\[ \xymatrix{ S_0 \ar[r] & S \ar[r] \ar@{-->}[rd] & A/I \\ R_0 \ar[r] \ar[u] & R \ar[r] \ar[u] & A \ar[u] } \]

with $R_0 \to S_0$ smooth, $R_0$ of finite type over $\mathbf{Z}$, and $S = S_0 \otimes _{R_0} R$. Let $x_1, \ldots , x_ n \in S_0$ be generators of $S_0$ over $R_0$. Let $a_1, \ldots , a_ n$ be elements of $A$ which map to the same elements in $A/I$ as the elements $x_1, \ldots , x_ n$. Denote $A_0 \subset A$ the subring generated by the image of $R_0$ and the elements $a_1, \ldots , a_ n$. Set $I_0 = A_0 \cap I$. Then $A_0/I_0 \subset A/I$ and $S_0 \to A/I$ maps into $A_0/I_0$. Thus it suffices to find the dotted arrow in the diagram

\[ \xymatrix{ S_0 \ar[r] \ar@{-->}[rd] & A_0/I_0 \\ R_0 \ar[r] \ar[u] & A_0 \ar[u] } \]

The ring $A_0$ is of finite type over $\mathbf{Z}$ by construction. Hence $A_0$ is Noetherian, whence $I_0$ is nilpotent, see Lemma 10.31.5. Say $I_0^ n = 0$. By Proposition 10.136.13 we can successively lift the $R_0$-algebra map $S_0 \to A_0/I_0$ to $S_0 \to A_0/I_0^2$, $S_0 \to A_0/I_0^3$, $\ldots $, and finally $S_0 \to A_0/I_0^ n = A_0$. $\square$

Your email address will not be published.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.

## Comments (0)