Fine Sheaves
A fine sheaf over X is one with "partitions of unity"; more precisely for any open cover of the space X we can find a family of homomorphisms from the sheaf to itself with sum 1 such that each homomorphism is 0 outside some element of the open cover.
Fine sheaves are usually only used over paracompact Hausdorff spaces X. Typical examples are the sheaf of continuous real functions over such a space, or smooth functions over a smooth (paracompact Hausdorff) manifold, or modules over these sheaves of rings.
Fine sheaves over paracompact Hausdorff spaces are soft and acyclic.
As an application, consider a real manifold X. There is the following resolution of the constant sheaf ℝ by the fine sheaves of (smooth) differential forms:
- 0 → ℝ → C0X → C1X → ... → Cdim XX → 0
This is a resolution, i.e. an exact complex of sheaves by the Poincaré lemma. The cohomology of X with values in ℝ can thus be computed as the cohomology of the complex of globally defined differential forms:
- Hi(X, ℝ) = Hi(C·X(X)).
Read more about this topic: Injective Sheaf
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