Lemma 31.27.3. Let $X$ be a locally Noetherian integral scheme. Let $\mathcal{L}$ be an invertible $\mathcal{O}_ X$-module. Let $s, s' \in \mathcal{K}_ X(\mathcal{L})$ be nonzero meromorphic sections of $\mathcal{L}$. Then $f = s/s'$ is an element of $R(X)^*$ and we have

$\sum \text{ord}_{Z, \mathcal{L}}(s)[Z] = \sum \text{ord}_{Z, \mathcal{L}}(s')[Z] + \text{div}(f)$

as Weil divisors.

Proof. This is clear from the definitions. Note that Lemma 31.27.2 guarantees that the sums are indeed Weil divisors. $\square$

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