---
date: 2021-10-21 18:42
modification date: Saturday 23rd October 2021 21:46:37
title: Chow ring
aliases: ["Chow group", "Chow groups", "Chow ring", "Chow rings", "Chow", rational equivalence]

---

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

---
- Tags: 
	- #AG 
- Refs:
	- #todo/add-references
- Links:
	- [computational properties of Chow](computational%20properties%20of%20Chow.md)

---

# Chow ring

- What is the **degree** of a cycle?

# Definitions

See [Weil divisor](Unsorted/Weil%20divisor.md).
![](attachments/Pasted%20image%2020220415134548.png)

Rational equivalence:
![](attachments/Pasted%20image%2020220415134755.png)

![](attachments/Pasted%20image%2020220415134849.png)

# Cycles associated to subschemes
![](attachments/Pasted%20image%2020220415134522.png)

Relation to [divisor class group](Unsorted/divisor%20class%20group.md):
![](attachments/Pasted%20image%2020220415135456.png)

## Paper on Chow Rings

- Reference: a recent result, [https://arxiv.org/pdf/math/0505560.pdf](https://arxiv.org/pdf/math/0505560.pdf)

Cohomology for [Symmetric algebra](Symmetric%20algebra) on the group of characters.

There is a map from the Chow ring back into cohomology, which in general fails surjectivity and injectivity. Tensoring this map with $\QQ$ creates an isomorphism, though. In this case, both have the ring structure of invariants under the [maximal torus](maximal%20torus). (Classical result, Leray and Borel.)

> An inverse of the [cycle class map](cycle%20class%20map)?

Chow rings have not been computed for $\PGL_n$. Need to know about [Euler class](Euler%20class.md),

$A_*$ known for all $O_n$ and $SO_n$ for $n$ odd in 80s, general result for $SO_n$ 2004. $PGL_n$ case is much harder. Understood for $n=2$, since $\PGL_2 \cong SO_3$. Other bits that have been computed: $H^*(\B\PGL_3, \ZZ/3), H^*(\B\PGL_n, \ZZ_2)$ for $n = 2 \mod 4$ in 70s/80s, incomplete results for $H^*(\B\PGL_p, \ZZ_p)$ in 2003.


Relation to [K-theory](Unsorted/K-theory.md):

$$\K_{0}(X) \otimes_{\mathbb{Z}} \mathbb{Q} \cong \prod_{i} \CH^{i}(X) \otimes_{\mathbb{Z}} \mathbb{Q}$$

When $X$ is a variety, $\CH_{\dim X - 1} X \cong \Div\Cl(X)$, the [divisor class group](Unsorted/divisor%20class%20group.md).
If $X\in \smooth\Var\slice k$, then $\CH_{\dim X - 1}X \cong \Pic(X)$, the [Picard group](Unsorted/Picard%20group.md).

# Products

![](attachments/Pasted%20image%2020220503120921.png)

# Relation to K theory

See [algebraic K theory](Unsorted/K-theory.md):
![](attachments/Pasted%20image%2020220503121545.png)

# Examples

![](attachments/Pasted%20image%2020220913171357.png)

![](attachments/Pasted%20image%2020220913171429.png)
![](attachments/Pasted%20image%2020220913171453.png)
![](attachments/Pasted%20image%2020220913171529.png)