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. $\square$

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