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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