Motives
Overview
Universal cohomology theory for algebraic varieties, proposed by Grothendieck. Should unify étale, de Rham, and Betti cohomology. The theory remains incomplete; mixed motives are an active area.
Era: 1960s–
Proposed at: Bures-sur-Yvette, France (IHES)
Formal structure
Domain: Smooth projective variety X over k; cycle groups Z^r(X)
M(X) = "universal" cohomology; H^*_ét ≃ H^*_dR ≃ H^*_Betti (conjectured)
Constraints: Chow groups; Künneth; weight filtration (mixed motives)
Objective: Unify cohomologies; motivic L-functions; standard conjectures