Lemma 42.37.2 (02U2)—The Stacks project

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Table of contents
Part 3: Topics in Scheme Theory
Chapter 42: Chow Homology and Chern Classes
Section 42.37: The Chern classes of a vector bundle
Lemma 42.37.2 (cite)

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Lemma 42.37.2. 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 42.37.1 is equal to the Weil divisor associated to $\mathcal{L}$ by Definition 42.24.1.

Proof. In this proof we use $c_1(\mathcal{L}) \cap [X]$ to denote the construction of Definition 42.24.1. 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 (42.37.1.1) 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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