Lemma 31.21.6. Let $Z \to Y \to X$ be immersions of schemes. Assume that $Z \to Y$ is $H_1$-regular. Then the canonical sequence of Morphisms, Lemma 29.31.5

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

is exact and locally split.

Proof. Since $\mathcal{C}_{Z/Y}$ is finite locally free (see Lemma 31.21.5 and Lemma 31.20.3) it suffices to prove that the sequence is exact. By what was proven in Morphisms, Lemma 29.31.5 it suffices to show that the first map is injective. Working affine locally this reduces to the following question: Suppose that we have a ring $A$ and ideals $I \subset J \subset A$. Assume that $J/I \subset A/I$ is generated by an $H_1$-regular sequence. Does this imply that $I/I^2 \otimes _ A A/J \to J/J^2$ is injective? Note that $I/I^2 \otimes _ A A/J = I/IJ$. Hence we are trying to prove that $I \cap J^2 = IJ$. This is the result of More on Algebra, Lemma 15.30.9. $\square$

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