Lemma 115.23.6 (02T0)—The Stacks project

Lemma 115.23.6. Let $(S, \delta )$ be as in Chow Homology, Situation 42.7.1. Let $X$ be locally of finite type over $S$. Assume $X$ integral with $\dim _\delta (X) = n$. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s$ be a nonzero meromorphic section of $\mathcal{L}$. Let $U \subset X$ be the maximal open subscheme such that $s$ corresponds to a section of $\mathcal{L}$ over $U$. There exists a projective morphism

\[ \pi : X' \longrightarrow X \]

such that

$X'$ is integral,
$\pi |_{\pi ^{-1}(U)} : \pi ^{-1}(U) \to U$ is an isomorphism,
there exist effective Cartier divisors $D, E \subset X'$ such that

\[ \pi ^*\mathcal{L} = \mathcal{O}_{X'}(D - E), \]
the meromorphic section $s$ corresponds, via the isomorphism above, to the meromorphic section $1_ D \otimes (1_ E)^{-1}$ (see Divisors, Definition 31.14.1),
we have

\[ \pi _*([D]_{n - 1} - [E]_{n - 1}) = \text{div}_\mathcal {L}(s) \]

in $Z_{n - 1}(X)$.

Proof. Let $\mathcal{I} \subset \mathcal{O}_ X$ be the quasi-coherent ideal sheaf of denominators of $s$, see Divisors, Definition 31.23.10. By Divisors, Lemma 31.34.6 we get (2), (3), and (4). By Divisors, Lemma 31.32.9 we get (1). By Divisors, Lemma 31.32.13 the morphism $\pi $ is projective. We still have to prove (5). By Chow Homology, Lemma 42.26.3 we have

\[ \pi _*(\text{div}_{\mathcal{L}'}(s')) = \text{div}_\mathcal {L}(s). \]

Hence it suffices to show that $\text{div}_{\mathcal{L}'}(s') = [D]_{n - 1} - [E]_{n - 1}$. This follows from the equality $s' = 1_ D \otimes 1_ E^{-1}$ and additivity, see Divisors, Lemma 31.27.5. $\square$

The Stacks project

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