Lemma 42.54.8. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $Z \subset X$ be a closed subscheme with virtual normal sheaf $\mathcal{N}$. Let $Y \to X$ be locally of finite type and $c \in A^*(Y \to X)$. Then $c$ and $c(Z \to X, \mathcal{N})$ commute (Remark 42.33.6).
Proof. To check this we may use Lemma 42.35.3. Thus we may assume $X$ is an integral scheme and we have to show $c \cap c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to X, \mathcal{N}) \cap c \cap [X]$ in $\mathop{\mathrm{CH}}\nolimits _*(Z \times _ X Y)$.
If $Z = X$, then $c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{N})$ by Lemma 42.54.5 which commutes with the bivariant class $c$, see Lemma 42.38.9.
Assume that $Z$ is not equal to $X$. By Lemma 42.35.3 it even suffices to prove the result after blowing up $X$ (in a nonzero ideal). Let us blowup $X$ in the ideal sheaf of $Z$. This reduces us to the case where $Z$ is an effective Cartier divisor, see Divisors, Lemma 31.32.4,
If $Z$ is an effective Cartier divisor, then we have
\[ c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ i^* \]
where $i^* \in A^1(Z \to X)$ is the gysin homomorphism associated to $i : Z \to X$ (Lemma 42.33.3) and $\mathcal{E}$ is the dual of the kernel of $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$, see Lemmas 42.54.3 and 42.54.7. Then we conclude because Chern classes are in the center of the bivariant ring (in the strong sense formulated in Lemma 42.38.9) and $c$ commutes with the gysin homomorphism $i^*$ by definition of bivariant classes. $\square$
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