---
date: 2022-01-15 21:49
modification date: Sunday 23rd January 2022 20:14:35
title: class group
aliases: ["ideal class group", "class group", "Picard group"]

---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- Summary of adelic class groups: <https://www2.math.upenn.edu/~yeya/class_group.pdf#page=1> #resources/summaries 
- Links:
	- #todo/create-links 

---

# Picard group

![](attachments/Pasted%20image%2020221030210451.png)

![](attachments/Pasted%20image%2020220914190226.png)

# Adelic class groups


![](attachments/Pasted%20image%2020220427092128.png)
![](attachments/Pasted%20image%2020220427092140.png)

The algebraic class group $\Cl(\GL_n(K))$ is a moduli of lattices in $K^n$.

# ideal class group


## For Noetherian domains

Write $\fracId(R)$ for the monoid of [fractional ideals](fractional%20ideals).

- For a Noetherian domain $R$: the **ideal group** $\fracId\units(R)$ is the abelian group of invertible [fractional ideals](Unsorted/fractional%20ideal.md).
  The elements are not necessarily ideals.
	  - Uses that $\fracId(R)$ is commutative and associative under multiplication with unit $R = \gens{1}$, so the subset of invertible elements forms a commutative group.
- Every *principal* fractional ideal is invertible using $\gens{x}\inv = \gens{x\inv}$, and they are closed under multiplication since $\gens{x}\gens{y} = \gens{xy}$, so the **principal fractional ideals** form a subgroup $\Prin\fracId(R) \leq \fracId\units(R)$.
  - The **ideal class group** or **Picard group** of $R$ is the quotient $\cl(R) \da \Pic(R) \da \fracId\units(R)/\Prin\fracId(R)$.

## For orders in number fields

- For $\OO\in\Ord(K)$, let $\fracId\units(\OO)$ be the invertible fractional ideals and $\Prin\fracId\units(\OO) \subseteq \fracId\units(\OO)$ be the principal fractional ideals of $\OO$. The quotient $\Cl(\OO) \da \fracId\units(\OO) / \Prin\fracId\units(\OO)$ is the **ideal class group** of the [order](Unsorted/order.md) $\OO$.
- There is a SES in $\mods{G_K}$: $$1 \to \Prin\fracId\units(\OO) \to \fracId\units(\OO) \to \Cl(\OO) \to 1,$$ and the corresponding LES starts $$
1 \to \Prin\fracId\units(\OO)^G \to \fracId\units(\OO)^G\to \Cl(\OO)^G \to H^1(\Prin\fracId\units(\OO)) \to 1
.$$

> Todo: not sure if this is actually for orders, or just for class groups of rings as in the previous section.

## Exercises

- Show that a DVR $R$ with a [uniformizer](uniformizer) has $\cl(R) \cong \ZZ$.
- Show that a [Dedekind domain](Unsorted/Dedekind%20domain.md) $R$ is a UFD iff $\cl(R) = 1$.
	- Show that if $R$ is an integrally closed Noetherian domain then $\cl(R) = 1$ when $R$ is a UFD.
	- Show that the converse holds if $\cl(R)$ is replaced with the [divisor class group](divisor%20class%20group).
	- Show that the ideal class group and [divisor class group](divisor%20class%20group.md) coincide for DVRs