The Stacks project

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

\[ \xymatrix{ Z' \ar[r] \ar[d]_ g & X' \ar[d]^ f \\ Z \ar[r] & X } \]

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

\[ \xymatrix{ g^*\mathcal{N}^\vee \ar[r] \ar[d] & (\mathcal{N}')^\vee \ar[d] \\ g^*\mathcal{C}_{Z/X} \ar[r] & \mathcal{C}_{Z'/X'} } \]

commutes. Then we have

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

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

Proof. By Lemma 42.54.2 we have $res(c(Z \to X, \mathcal{N})) = c(Z' \to X', g^*\mathcal{N})$ and the equality follows from Lemma 42.54.3. $\square$

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