Lemma 72.7.5 (0EPV)—The Stacks project

Lemma 72.7.5. Let $S$ be a scheme. Let $X$ be a locally Noetherian integral algebraic space over $S$ . Let $\mathcal{L}$ , $\mathcal{N}$ be invertible $\mathcal{O}_ X$ -modules. Let $s$ , resp. $t$ be a nonzero meromorphic section of $\mathcal{L}$ , resp. $\mathcal{N}$ . Then $st$ is a nonzero meromorphic section of $\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}$ and

$\text{div}_{\mathcal{L} \otimes \mathcal{N}}(st) = \text{div}_\mathcal {L}(s) + \text{div}_\mathcal {N}(t)$

in $\text{Div}(X)$ . In particular, the Weil divisor class of $\mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{N}$ is the sum of the Weil divisor classes of $\mathcal{L}$ and $\mathcal{N}$ .

Proof. Let $s$ , resp. $t$ be a nonzero meromorphic section of $\mathcal{L}$ , resp. $\mathcal{N}$ . Then $st$ is a nonzero meromorphic section of $\mathcal{L} \otimes \mathcal{N}$ . Let $Z \subset X$ be a prime divisor. Let $\xi \in |Z|$ be its generic point. Choose generators $s_\xi \in \mathcal{L}_\xi$ , and $t_\xi \in \mathcal{N}_\xi$ with notation as described earlier in this section. Then $s_\xi \otimes t_\xi$ is a generator for $(\mathcal{L} \otimes \mathcal{N})_\xi$ . So $st/(s_\xi t_\xi ) = (s/s_\xi )(t/t_\xi )$ in $Q(\mathcal{O}_{X, \xi }^ h)$ . Applying the additivity of Algebra, Lemma 10.121.1 we conclude that

$\text{div}_{\mathcal{L} \otimes \mathcal{N}, Z}(st) = \text{div}_{\mathcal{L}, Z}(s) + \text{div}_{\mathcal{N}, Z}(t)$

Some details omitted. $\square$

The Stacks project

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