---
date: 2022-04-08 10:02
modification date: Friday 8th April 2022 10:02:16
title: "Lefschetz motive"
aliases: [Lefschetz motive, Tate motive]
---

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

---
- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [derived stack](derived%20stack.md)
	- [K theory in AG](Unsorted/K%20theory%20in%20AG.md)
	- [weighted point count](weighted%20point%20count.md)

---

# Lefschetz motive

- $\LL = [\AA^1\slice k]$ is the **Lefschetz motive** and $\TT = \LL\inv$ is the **Tate motive**, its tensor inverse.


![](attachments/Pasted%20image%2020220408100217.png)
![](attachments/Pasted%20image%2020220408100238.png)

![](attachments/Pasted%20image%2020220420134919.png)

![](attachments/Pasted%20image%2020220529013737.png)

$$
\ZZ(1) \da M\ZZ_X \smashprod \Sigma^\infty (\PP^1, \ts{\infty})[-2] \qquad \in \DM(X)
$$