---
tag: 8WAC71N0C2
kind: definition
name: Weil divisor
msc: [14C20]
defines: Z9DN23DAE2
uses: [ND6XEY68P4, 17NVR4XBHK, X30G4V7VAJ, MGM3CNJW9V, 082GHNPJ53]
sources:
  - url: https://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)
---

## Definition

Let $X$ be a [[Noetherian|ND6XEY68P4]] [[integral|17NVR4XBHK]] [[separated|X30G4V7VAJ]] scheme that is regular in codimension one.
This means that the local ring of $X$ at every point of codimension one is a regular local ring of dimension one.
A *prime divisor* on $X$ is a closed integral subscheme $Y \subseteq X$ of codimension one.
A *Weil divisor* on $X$ is an element of the free [[abelian group|082GHNPJ53]] $\operatorname{Div}(X)$ generated by the prime divisors.
So it is a finite formal sum $D = \sum_i n_i Y_i$ with $n_i \in \Z$ and prime divisors $Y_i$.

## Remarks

Let $Y$ be a prime divisor with generic point $\eta_Y$.
The local ring $\mathcal{O}_{X, \eta_Y}$ is a [[discrete valuation ring|MGM3CNJW9V]], with valuation $v_Y$ on the function field $K$ of $X$.
For $f \in K^\times$, the *principal divisor* of $f$ is $\operatorname{div}(f) = \sum_Y v_Y(f) Y$, which is a finite sum.
On $\mathbb{P}^1_k$ over a field $k$, the function $f = x^2/(x - 1)$ has $\operatorname{div}(f) = 2[0] - [1] - [\infty]$.
A closed point of $\mathbb{A}^2_k$ is not a prime divisor, since it has codimension two, not one.
The quotient of $\operatorname{Div}(X)$ by the principal divisors is the divisor class group $\operatorname{Cl}(X)$.
For example, $\operatorname{Cl}(\mathbb{P}^n_k) \cong \Z$ by the degree, for $n \ge 1$.