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The Stacks project

Lemma 72.7.3. Let S be a scheme. Let X be a locally Noetherian integral algebraic space over S 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 72.7.2 guarantees that the sums are indeed Weil divisors. \square


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