Injective Sheaf - Fine Sheaves

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 → C0_X → C1_X → ... → Cdim X_X → 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)).