Serre Duality

A duality theorem for proper smooth n-dimensional algebraic or complex-analytic spaces giving a natural perfect pairing between the cohomology of a coherent sheaf and the cohomology of its dual twisted by the canonical (dualizing) sheaf, typically expressed as H^i(X,F) ≅ Hom(H^{n-i}(X, ω_X ⊗ F^∨), k)^* or an isomorphism H^i(X,F)^* ≅ H^{n-i}(X, ω_X ⊗ F^∨).