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• Tags
  • #motives
• Refs:
  • #todo/add-references
• Links:
  • Milnor Conjecture
  • SH
  • mixed motive
  • pure motive

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 \([{\mathbb{P}}^1] = [{\mathbb{A}}^1] + [{\mathbb{A}}^0]\) where \({\mathbb{A}}^0\coloneqq{\operatorname{pt}}\) and \([{\mathbb{P}}^2] = [{\mathbb{A}}^2] + [{\mathbb{A}}^1] + [{\mathbb{A}}^0]\).

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.

Intuition

From the transgressions in the Serre spectral sequence: