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Refs:
- 168p notes: https://arxiv.org/pdf/2004.06634.pdf#page=7
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motive
A conjectural geometric object associated to a smooth projective variety which is meant to capture the relationships between
- singular cohomology
- de Rham cohomology
- l-adic cohomology
- etale cohomology
- crystalline cohomology,
along with their relevant filtrations and comparison maps.
Meant to make sense of equations like \([{\mathbf{P}}^1] = [{\mathbf{A}}^1] + [{\mathbf{A}}^0]\) where \({\mathbf{A}}^0\coloneqq{\operatorname{pt}}\) and \([{\mathbf{P}}^2] = [{\mathbf{A}}^2] + [{\mathbf{A}}^1] + [{\mathbf{A}}^0]\).
Special motives:
- Chow motives
- Lefschetz motive
- Tate motive.
There is a conjecture tensor category of mixed motives where taking $\operatorname{Ext} $ recovers motivic cohomology, which should coincide with something predicted by algebraic K theory. Constructing this involves motivic homotopy theory.
Results condition on the Standard conjectures, Tate conjecture, and Hodge conjecture: 
Motivic invariants
# Intuition
From the transgressions in the Serre spectral sequence:
