Lemma 15.40.6. Let $A$ be a Noetherian complete local ring with residue field $k$. Let $B$ be a Noetherian complete local $k$-algebra. Assume $k \to B$ is formally smooth in the $\mathfrak m_ B$-adic topology. Then there exists a Noetherian complete local ring $C$ and a local homomorphism $A \to C$ which is formally smooth in the $\mathfrak m_ C$-adic topology such that $C \otimes _ A k \cong B$.

**Proof.**
Choose a diagram

as in Lemma 15.39.3. Let $t_1, \ldots , t_ d$ be a regular system of parameters for $R$ with $t_1 = p$ in case the characteristic of $k$ is $p > 0$. As $B$ and $\overline{S} = S \otimes _ R k$ are regular we see that $\mathop{\mathrm{Ker}}(\overline{S} \to B)$ is generated by elements $\overline{x}_1, \ldots , \overline{x}_ r$ which form part of a regular system of parameters of $\overline{S}$, see Algebra, Lemma 10.106.4. Lift these elements to $x_1, \ldots , x_ r \in S$. Then $t_1, \ldots , t_ d, x_1, \ldots , x_ r$ is part of a regular system of parameters for $S$. Hence $S/(x_1, \ldots , x_ r)$ is a power series ring over a field (if the characteristic of $k$ is zero) or a power series ring over a Cohen ring (if the characteristic of $k$ is $p > 0$), see Lemma 15.39.2. Moreover, it is still the case that $R \to S/(x_1, \ldots , x_ r)$ maps $t_1, \ldots , t_ d$ to a part of a regular system of parameters of $S/(x_1, \ldots , x_ r)$. In other words, we may replace $S$ by $S/(x_1, \ldots , x_ r)$ and assume we have a diagram

as in Lemma 15.39.3 with moreover $\overline{S} = B$. In this case $R \to S$ is formally smooth in the $\mathfrak m_ S$-adic topology by Proposition 15.40.5. Hence the base change $C = S \otimes _ R A$ is formally smooth over $A$ in the $\mathfrak m_ C$-adic topology by Lemma 15.37.8. $\square$

## 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Your email address will not be published.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.

## Comments (0)

There are also: