Lemma 81.28.3. In Situation 81.2.1 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 81.28.2 is equal to the bivariant class of Lemma 81.26.4.

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 81.27. Hence the equation in Lemma 81.28.1 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 81.28.2. Since $c_0^{new}(\mathcal{L}) = 1$ and $c_1(\mathcal{L})^0 = 1$ we conclude. $\square$

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