For a smooth projective -dimensional variety over a field and a coherent sheaf , Serre duality gives a perfect pairing between and , where is the canonical sheaf.
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Serre duality is a fundamental theoretical result in algebraic geometry and algebraic topology that relates cohomology groups of a projective variety, or a more general topological space, in a way that connects singular cohomology with dual spaces. Named after Jean-Pierre Serre, the duality provides a bridge between the geometry of a space and its cohomological properties.