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

Last modified date: <%+ tp.file.last_modified_date() %>

---
- Tags: 
	- #AG/basics 
- Refs:
	- See Bhatt-Lurie 2022 for [Cartier-Witt divisors](Cartier-Witt%20stack), generalized Cartier divisors, and their relation to [prisms](Unsorted/prismatic%20cohomology.md):
	  <https://arxiv.org/pdf/2201.06120.pdf#page=1> 
	  #resources/papers/2022 
- Links:
	- [divisor](Unsorted/divisor.md)
	- Ring theory:
		- [Krull dimension](Unsorted/Krull%20dimension.md)
		- [regular ring](Unsorted/regular%20ring.md)
	- Scheme theory:
		- [Noetherian scheme](Unsorted/Noetherian%20scheme.md)
		- [integral scheme](Unsorted/integral%20scheme.md)
		- [separated scheme](Unsorted/separated.md)
		- [closed subscheme](Unsorted/closed%20immersion.md)

---

# Summary

$\Cart\Div(X) \subseteq \Div(X)$ and it is interesting to ask when these are equal (i.e. when is an arbitrary divisor locally principal?)

![](attachments/Pasted%20image%2020220914182534.png)

- Idea:
	- **Weil divisors**: formal sums of hypersurfaces (codimension 1 subvarieties). Like homology, essentially $\CH_{n-1} = \CH^1$.
	- **Cartier divisors**: Weil divisors which are locally given by the divisor of a rational function, i.e. a locally principal divisor. Like cohomology.
		- Think of this as a formal $\ZZ\dash$combinations of embedded [integral](Unsorted/integrally%20closed.md) hypersurfaces.
		- Formal sums of arbitrary codimension subvarieties which can still be cut out by a single equation.

- Motivation:
	- If $X \subseteq \PP^n$ is codim 1 then $X=V(f)$ is a hypersurface cut out by a single equation. For arbitrary smooth varieties this may not hold, but $X$ is still locally a hypersurface. For singular varieties, even this can fail, so one needs to distinguish codimenson 1 subvarieties (~Weil) from those locally cut out by a single equation (~Cartier).
	  
# Facts

- Generally $\Cart\Div(X) \subseteq \operatorname{W}\Div(X)$. 
	- On smoooth varieties, Weil = Cartier.
![](attachments/Pasted%20image%2020220914183134.png)
A Weil divisor that is not Cartier:
![](attachments/Pasted%20image%2020220914185630.png)

# Weil divisor

- Related to the [divisor class group](Unsorted/divisor%20class%20group.md).

- For Weil divisors, only consider schemes which are [[noetherian]], [integral](Unsorted/integral%20scheme.md), [separated](Unsorted/separated.md), and [regular in codimension one](Unsorted/regular%20scheme.md).

- The **Weil divisor group of a scheme** is the free abelian group $\Div X$ generated by prime divisors, so finite sums 
  $$D \in \mathrm{WDiv}(X) \da \ZZ[\Sub^{n-1}(X)] \implies \quad D = \sum_{Y \in \Sub^{n-1}(X)} n_Y [Y].$$
	- A **prime divisor** on $X\in \Sch$ is a [closed](Unsorted/closed%20immersion.md) [integral](Unsorted/integral%20scheme.md) subscheme of codimension one (idea: irreducible subvarietes).
	- $D$ is **effective** iff $n_Y \geq 0$ for all $Y$.

- There is a map $\div: \OO_X \to \mathrm{WDiv}(X)$
- Every Weil divisor $D$ determines a coherent sheaf $\OO_X(D)$ where 
  $$\OO_X(D)(U) = \ts{f\in k(x) \st f = 0 \text{ or } \div(f) + D \geq 0 \text{ on } U},$$ leading to a SES 
  $$1 \to \OO_X(-D) \to \OO_X \to \OO_D \to 1.$$
  
# Cartier divisor

- A special type of Weil divisor. Related to the [Picard group](Unsorted/class%20group.md).
  
- The **Cartier divisor group of a variety** is defined as $\Cart\Div(X) \da \globsec{X; K_X\units/\OO_X\units}$ where $K_X$ is the sheaf of rational functions on $X$.
	- Equivalently, $\Cart\Div(X)$ is the group of [invertible](invertible%20ideals) [fractional ideals](Unsorted/fractional%20ideal.md).

- The **Cartier divisor group of a scheme** is the free abelian group generated by [closed subschemes](Unsorted/closed%20immersion.md) $D \subseteq X$ such that the [ideal sheaf](ideal%20sheaf.md) $\OO(-D) \subseteq \OO_X$ is [invertible](invertible) in $\mods{\OO_X}$.

- If $f\in K_X\units$ is a rational function on $X$, then the **divisor of $f$** is 
  $$(f) \da \Sum_{Y\in \Div(X) \, \mathrm{prime}} v_Y(f) Y.$$

- A Cartier divisor $D$ is **principal** iff $D = (f)$, the divisor of a function.
  
- Generally, $\operatorname{Pic}(X)=\Cart\Div(X) / \Prin\Cart\Div(X)$, i.e. the quotient of Cartier divisors by the principal divisors.
	- Equivalently, $\Pic(X) = \Cart\Div(X)/K\units$.

- Equivalently, a Cartier divisor is a pair $(\mcl, s \in \globsec{\mcl})$ with $s$ a rational section.
	- Effective when $s$ is everywhere defined, yielding a subscheme as $Z(s)$.

## Associating a line bundle to a Cartier divisor

![](attachments/Pasted%20image%2020220912142028.png)

# Results

- If $X$ is [factorial](factorial.md) (for instance, when $X$ is [smooth](smooth)), the Weil and Cartier divisor groups are isomorphic. 
- Cartier divisors do not behave well under base change: if $f\in \Sch(Y, X)$ and $D\subseteq X$, then $f\inv(D) \subseteq Y$ need not be be Cartier divisor.
	- For a more general notion, see Cartier-Witt divisors, e.g. in [Bhatt-Lurie 2022](https://arxiv.org/pdf/2201.06120.pdf#page=1)


# Arithmetic definitions of Cartier  and Weil divisors

![](attachments/Pasted%20image%2020220404011442.png)
![](attachments/Pasted%20image%2020220404011455.png)

# Examples and counterexamples

![](attachments/Pasted%20image%2020221226193219.png)

# Cartier-Witt divisors
![](attachments/Pasted%20image%2020220601201238.png)