Lemma 41.48.6. Let $(S, \delta )$ be as in Situation 41.7.1. 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

\[ c \circ c(Z \to X, \mathcal{N}) = c(Z \to X, \mathcal{N}) \circ c \]

in $A^*(Z \times _ X Y \to X)$.

**Proof.**
To check this we may use Lemma 41.31.12. Thus we may assume $X$ is an integral scheme and we have to show $c \cap c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to X, \mathcal{N}) \circ c \cap [X]$ in $A_*(Z \times _ X Y)$.

If $Z = X$, then $c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{N})$ by Lemma 41.48.4 which commutes with the bivariant class $c$, see Lemma 41.34.8.

Assume that $Z$ is not equal to $X$. By Lemma 41.31.12 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 30.32.4,

If $Z$ is an effective Cartier divisor, then we have

\[ c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{E}) \circ i^* \]

where $i^*$ is the gysin homomorphism associated to $i : Z \to X$ and $\mathcal{E}$ is the dual of the kernel of $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$, see Lemmas 41.48.3 and 41.48.5. Then we conclude because chern classes are in the center of the bivariant ring (in the strong sense formulated in Lemma 41.34.8) and commute with the gysin homomorphism $i^*$ by definition of bivariant classes.
$\square$

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