Sheaf cohomology assigns cohomology groups to a sheaf on a topological space by resolving the sheaf with the canonical Godement flasque resolution and taking the cohomology of the complex of global sections, which is needed because the global-section functor is only left exact. Learners see how flasque and fine sheaves supply acyclic resolutions, how hypercohomology extends the construction from a single sheaf to a complex of sheaves via the total cohomology of a double complex, and how the two spectral sequences of that double complex both converge to it. Comparing the two limits proves the de Rham theorem, its complex-coefficient version, the analytic de Rham theorem, and Dolbeault's theorem.
Four lectures (26 pages, CIMPA Summer School, Istanbul 2014) by Loring W. Tu that build sheaf cohomology from scratch, assuming only manifolds and the exterior calculus of differential forms. Lecture…