Lemma 42.54.3 (0FBL)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.54: Higher codimension gysin homomorphisms
Lemma 42.54.3 (cite)

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Lemma 42.54.3. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $\mathcal{N}$ be a virtual normal sheaf for a closed subscheme $Z$ of $X$. Suppose that we have a short exact sequence $0 \to \mathcal{N}' \to \mathcal{N} \to \mathcal{E} \to 0$ of finite locally free $\mathcal{O}_ Z$-modules such that the given surjection $\sigma : \mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ factors through a map $\sigma ' : (\mathcal{N}')^\vee \to \mathcal{C}_{Z/X}$. Then

\[ c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ c(Z \to X, \mathcal{N}') \]

as bivariant classes.

Proof. Denote $N' \to N$ the closed immersion of vector bundles corresponding to the surjection $\mathcal{N}^\vee \to (\mathcal{N}')^\vee $. Then we have closed immersions

\[ C_ ZX \to N' \to N \]

Thus the desired relationship between the bivariant classes follows immediately from Lemma 42.44.2. $\square$

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