6 min read•august 20, 2024
and are two ways to study sheaves on topological spaces. They use different approaches but are closely related. For paracompact Hausdorff spaces, they give the same results.
These cohomology theories help us understand global properties of sheaves by looking at local information. They're useful in algebraic geometry and complex analysis for classifying vector bundles and studying complex manifolds.
Sheaf cohomology satisfies a long exact sequence, which relates the cohomology groups of a short exact sequence of sheaves
Given a short exact sequence of sheaves , there is an induced long exact sequence in cohomology:
The long exact sequence is a powerful tool for computing sheaf cohomology groups and studying the relationships between sheaves