Lemma 72.7.3 (0EPT)—The Stacks project

Table of contents
Part 4: Algebraic Spaces
Chapter 72: Algebraic Spaces over Fields
Section 72.7: The Weil divisor class associated to an invertible module
Lemma 72.7.3 (cite)

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. $\square$

The Stacks project

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