---
date: 2022-04-10 18:43
modification date: Sunday 10th April 2022 18:43:34
title: "K theory in AG"
aliases: [K theory in AG]
---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# K theory in AG

Cohomological: vector bundles.

![](attachments/Pasted%20image%2020220410184336.png)

Homological: coherent sheaves.
![](attachments/Pasted%20image%2020220410184349.png)

Pushforwards:
![](attachments/Pasted%20image%2020220410184418.png)

![](attachments/Pasted%20image%2020220410184436.png)

Relation to [topological K theory](Unsorted/topological%20K%20theory.md):
![](attachments/Pasted%20image%2020220410184512.png)

![](attachments/Pasted%20image%2020220410185403.png)
![](attachments/Pasted%20image%2020220410185430.png)

# Motives

In $\K_0(\St\slice k)$,

- $\id = \ts{\spec k}$
- $\LL \da \ts{\AA^1\slice k}$ is the [Lefschetz motive](Unsorted/Lefschetz%20motive.md)
- If $X\to Y$ is a $G\dash$torsor, where $G$ is *special* in the sense that every $G\dash$torsor is Zariski-locally trivial, then $\ts{X} = \ts{G}\cdot \ts{Y}$.
	- $\ts{G}\inv = \ts{\BG}$ using that $\spec k \to \BG$ is the universal torsor.
	- If $X\to Y$ is a $\GG_m\dash$torsor of finite-type algebraic stacks, $[Y] = [X] \cdot [\GG_m]\inv$.
- $\ts{\GG_m} = \ts{\GL_1} = \LL$.
- $\ts{\GL_n} = \prod_{1\leq i\leq n}(\LL^n - \LL^i)$
- $\ts{\SL_n} = (\LL-1)\inv \ts{\GL_n}$ since $\GL_n \to \GG_m$ is an $\SL_n\dash$torsor, so $\ts{\SL_n} = \ts{\GL_n}\cdot \ts{\GG_m}\inv$.
- $\ts{\SL_2} = \LL(\LL^2-1)$.
- Since $\GL_2\to \PGL_2$ is a $\GG_m\dash$torsor, $\ts{\PGL_n} = \ts{\GL_2} \cdot \ts{\GG_m}\inv$.
- $\ts{\PGL_2}\inv = \ts{\B \PGL_2} = \qty{\LL(\LL^2-1)}\inv$.

![](attachments/Pasted%20image%2020220422230212.png)