Chapter 5 - Cohomology

Summary

We now wish to measure the lack of exactness of the global section functor Γ(X, -); we have seen that it is left exact, but need not take a sheaf epimorphism into a surjective map of sections.

We first consider the problem in the general setting of homological algebra: we wish to mend the lack of right exactness of a left exact functor between abelian categories. This leads us to define injective objects, and to show that they can be used to define the right derived functors of our functor, which fit into a long exact sequence extending the left exact sequence it produces. The right derived functors have a suitable universal property, which is used to obtain identities concerning composite functors.

We next apply this procedure to the case of sheaves. Having verified that there are enough injective sheaves, we deduce the existence of cohomology functors fitting into a long exact sequence. The general method also yields the higher direct images of a morphism, which generalise the cohomology groups, but may be expressed in terms of them. We investigate the processes of changing structure sheaves and base rings, and summarise an alternative approach to this universal cohomology theory, using flasque sheaves.