Lemma 31.27.3 (02SH)—The Stacks project

Processing math: 100%

Table of contents
Part 2: Schemes
Chapter 31: Divisors
Section 31.27: The Weil divisor class associated to an invertible module
Lemma 31.27.3 (cite)

previous lemma
next definition

numberstags

history
statistics
2 tags refer to this tag

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. $\square$

Comments (0)

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$ ). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Comments (0)

Post a comment