Lemma 42.36.2. Let $(S, \delta )$ be as in Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ is integral and $n = \dim _\delta (X)$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. The first Chern class of $\mathcal{L}$ on $X$ of Definition 42.36.1 is equal to the Weil divisor associated to $\mathcal{L}$ by Definition 42.23.1.

Proof. In this proof we use $c_1(\mathcal{L}) \cap [X]$ to denote the construction of Definition 42.23.1. Since $\mathcal{L}$ has rank $1$ we have $\mathbf{P}(\mathcal{L}) = X$ and $\mathcal{O}_{\mathbf{P}(\mathcal{L})}(1) = \mathcal{L}$ by our normalizations. Hence (42.36.1.1) reads

$(-1)^1 c_1(\mathcal{L}) \cap c_0 + (-1)^0 c_1 = 0$

Since $c_0 = [X]$, we conclude $c_1 = c_1(\mathcal{L}) \cap [X]$ as desired. $\square$

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