Lemma 10.140.4. Let $k$ be a field. Let $R$ be a Noetherian local ring containing $k$. Assume that the residue field $\kappa = R/\mathfrak m$ is a finitely generated separable extension of $k$. Then the map

is injective.

Lemma 10.140.4. Let $k$ be a field. Let $R$ be a Noetherian local ring containing $k$. Assume that the residue field $\kappa = R/\mathfrak m$ is a finitely generated separable extension of $k$. Then the map

\[ \text{d} : \mathfrak m/\mathfrak m^2 \longrightarrow \Omega _{R/k} \otimes _ R \kappa (\mathfrak m) \]

is injective.

**Proof.**
We may replace $R$ by $R/\mathfrak m^2$. Hence we may assume that $\mathfrak m^2 = 0$. By assumption we may write $\kappa = k(\overline{x}_1, \ldots , \overline{x}_ r, \overline{y})$ where $\overline{x}_1, \ldots , \overline{x}_ r$ is a transcendence basis of $\kappa $ over $k$ and $\overline{y}$ is separable algebraic over $k(\overline{x}_1, \ldots , \overline{x}_ r)$. Say its minimal equation is $P(\overline{y}) = 0$ with $P(T) = T^ d + \sum _{i < d} a_ iT^ i$, with $a_ i \in k(\overline{x}_1, \ldots , \overline{x}_ r)$ and $P'(\overline{y}) \not= 0$. Choose any lifts $x_ i \in R$ of the elements $\overline{x}_ i \in \kappa $. This gives a commutative diagram

\[ \xymatrix{ R \ar[r] & \kappa \\ & k(\overline{x}_1, \ldots , \overline{x}_ r) \ar[lu]^\varphi \ar[u] } \]

of $k$-algebras. We want to extend the left upwards arrow $\varphi $ to a $k$-algebra map from $\kappa $ to $R$. To do this choose any $y \in R$ lifting $\overline{y}$. To see that it defines a $k$-algebra map defined on $\kappa \cong k(\overline{x}_1, \ldots , \overline{x}_ r)[T]/(P)$ all we have to show is that we may choose $y$ such that $P^\varphi (y) = 0$. If not then we compute for $\delta \in \mathfrak m$ that

\[ P(y + \delta ) = P(y) + P'(y)\delta \]

because $\mathfrak m^2 = 0$. Since $P'(y)\delta = P'(\overline{y})\delta $ we see that we can adjust our choice as desired. This shows that $R \cong \kappa \oplus \mathfrak m$ as $k$-algebras! Now either a direct computation of $\Omega _{\kappa \oplus \mathfrak m/k}$ or an application of Lemma 10.131.10 finishes the proof. $\square$

Your email address will not be published.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 (2)

Comment #8331 by Et on

Comment #8946 by Stacks project on