The Stacks project

Lemma 42.54.7. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be a scheme locally of finite type over $S$. Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in Definition 42.29.1. The gysin homomorphism $i^*$ viewed as an element of $A^1(D \to X)$ (see Lemma 42.33.3) is the same as the bivariant class $c(D \to X, \mathcal{N}) \in A^1(D \to X)$ constructed using $\mathcal{N} = i^*\mathcal{L}$ viewed as a virtual normal sheaf for $D$ in $X$.

Proof. We will use the criterion of Lemma 42.35.3. Thus we may assume that $X$ is an integral scheme and we have to show that $i^*[X]$ is equal to $c \cap [X]$. Let $n = \dim _\delta (X)$. As usual, there are two cases.

If $X = D$, then we see that both classes are represented by $c_1(\mathcal{N}) \cap [X]_ n$. See Lemma 42.54.5 and Definition 42.29.1.

If $D \not= X$, then $D \to X$ is an effective Cartier divisor and in particular a regular closed immersion of codimension $1$. Again by Lemma 42.54.5 we conclude $c(D \to X, \mathcal{N}) \cap [X]_ n = [D]_{n - 1}$. The same is true by definition for the gysin homomorphism and we conclude once again. $\square$

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