Lemma 31.27.5. Let $X$ be a locally Noetherian integral scheme. 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 $\mathcal{L} \otimes \mathcal{N}$, 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 $. Then $s_\xi t_\xi $ is a generator for $(\mathcal{L} \otimes \mathcal{N})_\xi $. So $st/(s_\xi t_\xi ) = (s/s_\xi )(t/t_\xi )$. Hence we see that

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

by the additivity of the $\text{ord}_ Z$ function.
$\square$

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