The Stacks project

Lemma 15.40.4. Let $A \to B$ be a local homomorphism of Noetherian local rings. Assume $A \to B$ is formally smooth in the $\mathfrak m_ B$-adic topology. Let $K$ be the residue field of $B$. Then the Jacobi-Zariski sequence for $A \to B \to K$ gives an exact sequence

\[ 0 \to H_1(\mathop{N\! L}\nolimits _{K/A}) \to \mathfrak m_ B/\mathfrak m_ B^2 \to \Omega _{B/A} \otimes _ B K \to \Omega _{K/A} \to 0 \]

Proof. Observe that $\mathfrak m_ B/\mathfrak m_ B^2 = H_1(\mathop{N\! L}\nolimits _{K/B})$ by Algebra, Lemma 10.134.6. By Algebra, Lemma 10.134.4 it remains to show injectivity of $H_1(\mathop{N\! L}\nolimits _{K/A}) \to \mathfrak m_ B/\mathfrak m_ B^2$. With $k$ the residue field of $A$, the Jacobi-Zariski sequence for $A \to k \to K$ gives $\Omega _{K/A} = \Omega _{K/k}$ and an exact sequence

\[ \mathfrak m_ A/\mathfrak m_ A^2 \otimes _ k K \to H_1(\mathop{N\! L}\nolimits _{K/A}) \to H_1(\mathop{N\! L}\nolimits _{K/k}) \to 0 \]

Set $\overline{B} = B \otimes _ A k$. Since $\overline{B}$ is regular the ideal $\mathfrak m_{\overline{B}}$ is generated by a regular sequence. Applying Lemmas 15.30.9 and 15.30.7 to $\mathfrak m_ A B \subset \mathfrak m_ B$ we find $\mathfrak m_ A B / (\mathfrak m_ AB \cap \mathfrak m_ B^2) = \mathfrak m_ A B / \mathfrak m_ A \mathfrak m_ B$ which is equal to $\mathfrak m_ A/\mathfrak m_ A^2 \otimes _ k K$ as $A \to B$ is flat by Lemma 15.40.3. Thus we obtain a short exact sequence

\[ 0 \to \mathfrak m_ A/\mathfrak m_ A^2 \otimes _ k K \to \mathfrak m_ B/\mathfrak m_ B^2 \to \mathfrak m_{\overline{B}}/\mathfrak m_{\overline{B}}^2 \to 0 \]

Functoriality of the Jacobi-Zariski sequences shows that we obtain a commutative diagram

\[ \xymatrix{ & \mathfrak m_ A/\mathfrak m_ A^2 \otimes _ k K \ar[d] \ar[r] & H_1(\mathop{N\! L}\nolimits _{K/A}) \ar[d] \ar[r] & H_1(\mathop{N\! L}\nolimits _{K/k}) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \mathfrak m_ A/\mathfrak m_ A^2 \otimes _ k K \ar[r] & \mathfrak m_ B/\mathfrak m_ B^2 \ar[r] & \mathfrak m_{\overline{B}}/\mathfrak m_{\overline{B}}^2 \ar[r] & 0 } \]

The left vertical arrow is injective by Theorem 15.40.1 as $k \to \overline{B}$ is formally smooth in the $\mathfrak m_{\overline{B}}$-adic topology by Lemma 15.37.8. This finishes the proof by the snake lemma. $\square$


Comments (0)

There are also:

  • 3 comment(s) on Section 15.40: Geometric regularity and formal smoothness

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Your email address will not be published.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