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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