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I am reading about Serre duality theory in algebraic geometry from Hartshorne, and am wondering what kinds of applications it has. It seems that most applications go through some version of the Riemann-Roch theorem, which is then used to study algebraic curves and surfaces.

What are some other applications of Serre duality?

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My favorite is the following. Serre duality shows that the derived category of a (smooth) algebraic variety has a Serre functor, which is not just an invariant of the derived category but it also produces left (right) adjoint to a functor which has right (left) adjoint.

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Serre functors are an extremely powerful tool in the study of derived categories of coherent sheaves, see for some references and more information. – Adeel Jul 29 '13 at 18:40

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