Lemma 82.28.3. In Situation 82.2.1 let X/B be good. Let \mathcal{L} be an invertible \mathcal{O}_ X-module. The first Chern class of \mathcal{L} on X of Definition 82.28.2 is equal to the bivariant class of Lemma 82.26.4.
Proof. Namely, in this case P = \mathbf{P}(\mathcal{L}) = X with \mathcal{O}_ P(1) = \mathcal{L} by our normalization of the projective bundle, see Section 82.27. Hence the equation in Lemma 82.28.1 reads
(-1)^0 c_1(\mathcal{L})^0 \cap c^{new}_1(\mathcal{L}) \cap \alpha + (-1)^1 c_1(\mathcal{L})^1 \cap c^{new}_0(\mathcal{L}) \cap \alpha = 0
where c_ i^{new}(\mathcal{L}) is as in Definition 82.28.2. Since c_0^{new}(\mathcal{L}) = 1 and c_1(\mathcal{L})^0 = 1 we conclude. \square
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