---
tag: JX3ZH5WY0T
kind: definition
name: Cartier divisor
msc: [14C20, 14C22]
defines: FHV05ZK99F
uses: [5A35TYZD02, TB6TM6R4MD, K5Z67HR9ET, 07C6ECNJET, 049PXYHW1T]
sources:
  - url: https://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)
---

## Definition

Let $X$ be a [[scheme|5A35TYZD02]] with structure sheaf $\mathcal{O}_X$.
For an open set $U \subseteq X$, let $S(U)$ be the set of $s \in \mathcal{O}_X(U)$ whose germ $s_x$ is a non-zero-divisor in $\mathcal{O}_{X,x}$ for every $x \in U$.
The *sheaf of total quotient rings* $\mathcal{K}_X$ is the [[sheaf|TB6TM6R4MD]] associated to the presheaf $U \mapsto S(U)^{-1}\mathcal{O}_X(U)$.
For an integral scheme it is the constant sheaf of the function field; in general it is built from the [[total ring of fractions|K5Z67HR9ET]].
Let $\mathcal{K}_X^*$ and $\mathcal{O}_X^*$ be the sheaves of [[units|07C6ECNJET]] of $\mathcal{K}_X$ and of $\mathcal{O}_X$.
A *Cartier divisor* on $X$ is a global section of the quotient sheaf $\mathcal{K}_X^* / \mathcal{O}_X^*$.

Equivalently, a Cartier divisor is given by an [[open cover|049PXYHW1T]] $(U_i)$ of $X$ and sections $f_i \in \mathcal{K}_X^*(U_i)$ such that $f_i / f_j \in \mathcal{O}_X^*(U_i \cap U_j)$ for all $i, j$.
Two such families $(U_i, f_i)$ and $(V_k, g_k)$ define the same divisor when $f_i / g_k \in \mathcal{O}_X^*(U_i \cap V_k)$ for all $i, k$.

## Remarks

The Cartier divisors form an abelian group, written additively, with $(U_i, f_i) + (U_i, g_i) = (U_i, f_i g_i)$.
A divisor is *principal* if it is given by one global $f \in \mathcal{K}_X^*(X)$, and *effective* if all $f_i$ lie in $\mathcal{O}_X(U_i)$.

Example: on $X = \mathbb{A}^1_k = \operatorname{Spec} k[t]$, the single pair $(X, t)$ is an effective principal Cartier divisor, the point $0$.
The compatibility condition holds because there is only one open set, and $t/t = 1$ is a unit.

Non-example: on $\mathbb{A}^1_k$ take $U = X$ with $f_U = t$ and $V = X \setminus \set{0}$ with $f_V = t - 1$.
On $U \cap V = V$, the ratio $t/(t-1)$ is not a unit, since it has a pole at $t = 1$, so the compatibility condition fails.

The divisor $D = (U_i, f_i)$ defines the invertible subsheaf $\mathcal{O}_X(D) \subseteq \mathcal{K}_X$ generated by $f_i^{-1}$ on $U_i$.