Lemma 10.139.1. Given ring maps $A \to B \to C$ with $B \to C$ smooth, then the sequence

of Lemma 10.131.7 is exact.

Lemma 10.139.1. Given ring maps $A \to B \to C$ with $B \to C$ smooth, then the sequence

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

of Lemma 10.131.7 is exact.

**Proof.**
This follows from the more general Lemma 10.138.9 because a smooth ring map is formally smooth, see Proposition 10.138.13. But it also follows directly from Lemma 10.134.4 since $H_1(L_{C/B}) = 0$ is part of the definition of smoothness of $B \to C$.
$\square$

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