Lemma 29.32.18. Let

be a commutative diagram of schemes where $i$ and $j$ are immersions and $X \to Y$ is smooth. Then the canonical exact sequence

of Lemma 29.31.18 is exact.

Lemma 29.32.18. Let

\[ \xymatrix{ Z \ar[r]_ i \ar[rd]_ j & X \ar[d] \\ & Y } \]

be a commutative diagram of schemes where $i$ and $j$ are immersions and $X \to Y$ is smooth. Then the canonical exact sequence

\[ 0 \to \mathcal{C}_{Z/Y} \to \mathcal{C}_{Z/X} \to i^*\Omega _{X/Y} \to 0 \]

of Lemma 29.31.18 is exact.

**Proof.**
The algebraic version of this lemma is the following: Given ring maps $A \to B \to C$ with $A \to C$ surjective and $A \to B$ smooth, then the sequence

\[ 0 \to I/I^2 \to J/J^2 \to C \otimes _ B \Omega _{B/A} \to 0 \]

of Algebra, Lemma 10.133.7 is exact. This is Algebra, Lemma 10.138.3. $\square$

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