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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