Lemma 29.31.5. Let $Z \to Y \to X$ be immersions of schemes. Then there is a canonical exact sequence

$i^*\mathcal{C}_{Y/X} \to \mathcal{C}_{Z/X} \to \mathcal{C}_{Z/Y} \to 0$

where the maps come from Lemma 29.31.3 and $i : Z \to Y$ is the first morphism.

Proof. Via Lemma 29.31.3 this translates into the following algebra fact. Suppose that $C \to B \to A$ are surjective ring maps. Let $I = \mathop{\mathrm{Ker}}(B \to A)$, $J = \mathop{\mathrm{Ker}}(C \to A)$ and $K = \mathop{\mathrm{Ker}}(C \to B)$. Then there is an exact sequence

$K/K^2 \otimes _ B A \to J/J^2 \to I/I^2 \to 0.$

This follows immediately from the observation that $I = J/K$. $\square$

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