Sheaf cohomology
Sheaf cohomology is a powerful tool in algebraic geometry which unfortunately takes quite a bit of abstract machinery to set up properly. However, once the basic theory is set up, one can often work with cohomology without revisiting the abstract machinery. In this video, we present this basic theory which allows algebraic geometers to work with the cohomology of coherent sheaves on projective varieties. In particular, we touch on Grothendieck vanishing, the relation with Cech cohomology, Serre vanishing and Serre finiteness. Cohomology can be seen as a machine to generate lots of good numerical invariants. As an example, we see how the degree of the Hilbert-Serre polynomial gives the dimension of a sheaf.