Lemma 42.54.4. Let $(S, \delta )$ be as in Situation 42.7.1. Consider a cartesian diagram

of schemes locally of finite type over $S$ whose horizontal arrows are closed immersions. Let $\mathcal{N}$, resp. $\mathcal{N}'$ be a virtual normal sheaf for $Z \subset X$, resp. $Z' \to X'$. Assume given a short exact sequence $0 \to \mathcal{N}' \to g^*\mathcal{N} \to \mathcal{E} \to 0$ of finite locally free modules on $Z'$ such that the diagram

commutes. Then we have

in $A^*(Z' \to X')^\wedge $.

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