Étale cohomology
Overview
Cohomology for schemes using étale topology. Replaces singular cohomology in positive characteristic. Deligne (1974) proved Weil conjectures with it.
Era: 1960s–1974
Proposed at: Bures-sur-Yvette, France (IHES)
Formal structure
Domain: Scheme X; étale sheaf F; ℓ-adic coefficients (ℓ ≠ p)
H^i_ét(X, F) = lim H^i(X_ét, F/ℓ^n); H^*(X) = ⊕ H^i
Constraints: Finiteness (proper X); Poincaré duality; Lefschetz fixed-point
Objective: Weil conjectures; ℓ-adic representations; L-functions