Serre duality and the adjunction formula

Serre duality is one of the fundamental tools in algebraic geometry. In this video, we explore this theorem for smooth, or more generally, Cohen-Macaulay projective varieties of arbitrary dimension. As usual, it relates cohomology groups to certain Ext groups involving the so-called dualising sheaf which , in the case of a smooth projective variety is just the canonical sheaf. It thus helps give effective bounds for ensuring certain cohomology groups vanish, as we shall see. We also give an adjunction formula which computes the dualising sheaf of an irreducible Cartier divisor on a smooth variety. We see how this generalises the adjunction formula we saw earlier in this playlist for plane curves. We finish by sketching the proof of the Serre duality theorem which involves first proving it for projective space, and then bootstrapping this result to prove it for other projective varieties by proving various adjunction formulae.