Lemma 42.59.1 (0FF0)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.59: Gysin maps for local complete intersection morphisms
Lemma 42.59.1 (cite)

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Lemma 42.59.1. Let $(S, \delta )$ be as in Situation 42.7.1. Let $i : X \to Y$ and $j : Y \to Z$ be regular immersions of schemes locally of finite type over $S$ . Then $j \circ i$ is a regular immersion and $(j \circ i)^! = i^! \circ j^!$ .

Proof. The first statement is Divisors, Lemma 31.21.7. By Divisors, Lemma 31.21.6 there is a short exact sequence

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

Thus the result by the more general Lemma 42.54.10. $\square$

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