---
date: 2022-03-29 17:19
modification date: Tuesday 29th March 2022 17:19:12
title: "motive"
aliases: [motive, motives, motivic]

---

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


---

- Tags 
	- #motivic 
- Refs:
	- 168p notes: <https://arxiv.org/pdf/2004.06634.pdf#page=7>
- Links:
	- [Milnor Conjecture](Unsorted/Milnor%20Conjecture.md)
	- [SH](SH.md)
	- [Unsorted/mixed motive](Unsorted/mixed%20motive.md)
	- [Unsorted/pure motive](Unsorted/pure%20motive.md)
	- [motivic homotopy](Unsorted/motivic%20homotopy.md)
	- [[the category of motives]]

---

# motive

![](attachments/2023-03-08constr.png)

A conjectural geometric object associated to a smooth projective variety which is meant to capture the relationships between

- [singular cohomology](singular%20cohomology)
- [de Rham cohomology](Unsorted/algebraic%20de%20Rham%20cohomology.md)
- [l-adic cohomology](Unsorted/l-adic%20cohomology.md)
- [etale cohomology](Unsorted/l-adic%20cohomology.md)
- [crystalline cohomology](Unsorted/crystalline%20cohomology.md),

along with their relevant filtrations and comparison maps.

![](attachments/Pasted%20image%2020220515020128.png)


Meant to make sense of equations like $[\PP^1] = [\AA^1] + [\AA^0]$  where $\AA^0\da \pt$ and $[\PP^2] = [\AA^2] + [\AA^1] + [\AA^0]$.

Special motives: 

- [Chow motives](Chow%20motives)
- [Lefschetz motive](Lefschetz%20motive.md) 
- [Tate motive](Tate%20motive).

There is a conjecture [tensor category](tensor%20category.md) of [mixed motives](mixed%20motives.md) where taking $\Ext$ recovers [motivic cohomology](Unsorted/motivic%20cohomology.md), which should coincide with something predicted by [algebraic K theory](Unsorted/K-theory.md). Constructing this involves [motivic homotopy theory](Unsorted/motivic%20homotopy.md).

Results condition on the [Standard conjectures](Standard%20conjectures), [Tate conjecture](Unsorted/tate%20conjecture.md), and [Hodge conjecture](Hodge%20conjecture):
![](attachments/Pasted%20image%2020220515020219.png)

# Motivic invariants

![](attachments/Pasted%20image%2020220410185024.png)
![](attachments/Pasted%20image%2020220410185049.png)
![](attachments/Pasted%20image%2020220410185102.png)
# Intuition

![](attachments/Pasted%20image%2020220410215743.png)
![](attachments/Pasted%20image%2020220410215928.png)
![](attachments/Pasted%20image%2020220410215944.png)

From the [transgressions in the Serre spectral sequence](Unsorted/Serre%20spectral%20sequence#Transgressions):

![](attachments/Pasted%20image%2020220403192047.png)
![](attachments/Pasted%20image%2020220403192059.png)

![](attachments/Pasted%20image%2020220410224855.png)
![](attachments/Pasted%20image%2020220410224909.png)
![](attachments/Pasted%20image%2020220410224922.png)
![](attachments/Pasted%20image%2020220410225017.png)