Lemma 42.53.3. 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

as bivariant classes.

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