Section 3.2 - Cohomology of Sheaves

In this note we define cohomology of sheaves by taking the derived functors of the global section functor. As an application of general techniques of cohomology we prove the Grothendieck and Serre vanishing theorems. We introduce the Čech cohomology and use it to calculate cohomology of projective space. The original reference for this material is EGA III, but most graduate students would probably encounter it in Hartshorne’s book [Har77] where many proofs are given only for noetherian schemes, probably because the only known proofs in the general case utilised spectral sequences. Several years after Hartshorne’s book was published there appeared a paper by Kempf [Kem80] giving very elegant and elementary proofs in the full generality of quasi-compact quasi-separated schemes. The proofs given here are a mix of those from Hartshorne’s book and Kempf’s paper.