Section 3.7 - The Serre Duality Theorem

In this note we prove the Serre duality theorem for the cohomology of coherent sheaves on a projective scheme. First we do the case of projective space itself. Then on an arbitrary projective scheme X, we show that there is a coherent sheaf ω◦ X , which plays the role in duality theory similar to the canonical sheaf of a nonsingular variety. In particular, if X is Cohen-Macaulay, it gives a duality theorem just like the one on projective space. Finally, if X is a nonsingular variety over an algebraically closed field we show that the dualising sheaf ω◦ X agrees with the canonical sheaf ωX .