Lemma 10.131.7. Let $A \to B \to C$ be ring maps. Then there is a canonical exact sequence

of $C$-modules.

Lemma 10.131.7. Let $A \to B \to C$ be ring maps. Then there is a canonical exact sequence

\[ C \otimes _ B \Omega _{B/A} \to \Omega _{C/A} \to \Omega _{C/B} \to 0 \]

of $C$-modules.

**Proof.**
We get a diagram (10.131.5.1) by putting $R = A$, $S = C$, $R' = B$, and $S' = C$. By Lemma 10.131.6 the map $\Omega _{C/A} \to \Omega _{C/B}$ is surjective, and the kernel is generated by the elements $\text{d}(c)$, where $c \in C$ is in the image of $B \to C$. The lemma follows.
$\square$

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