---
date: 2022-04-05 23:42
modification date: Tuesday 5th April 2022 23:42:25
title: "principal divisor"
aliases: [principal divisor]
---

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# principal divisor

Let $X$ be a noetherian integral separated scheme which is regular in codimension one and let $Y$ be a prime divisor with generic point $\eta$.
Then $\OO_{X, \eta}$ is a [DVR](Unsorted/DVR.md) with residue field $K$. 
There is a map
$$
K\units \to \Div(X) \\
f \mapsto (f) \da \sum_{y\in \Div(X)} v_y(f)\, y
,$$
which is well-defined since infinitely many $v_y(f) = 0$ since the non-regular locus of $f$ is a proper closed subset of a Noetherian scheme, and thus contains only finitely many [prime divisors](Unsorted/Weil%20divisor.md).
Any divisor in the image is called **prinicipal**.
![](attachments/Pasted%20image%2020220214091709.png)