(Soft and fine sheaves)
Let \(X\) be a paracompact Hausdorff space and \(\mathcal{F}\) a sheaf of abelian groups on \(X\).
\(\mathcal{F}\) is soft if for every closed \(Z \subseteq X\) the restriction \(\Gamma(X, \mathcal{F}) \to \Gamma(Z, \mathcal{F}|_Z)\) is surjective.
\(\mathcal{F}\) is c-soft if for every compact \(K \subseteq X\) the restriction \(\Gamma(X, \mathcal{F}) \to \Gamma(K, \mathcal{F}|_K)\) is surjective.
\(\mathcal{F}\) is fine if for every locally finite open cover \(\{U_i\}\) of \(X\) there are sheaf endomorphisms \(\eta_i \colon \mathcal{F} \to \mathcal{F}\) with the support of \(\eta_i\) contained in \(U_i\) and \(\sum_i \eta_i = \mathrm{id}_{\mathcal{F}}\).
On a paracompact Hausdorff space, flasque sheaves and fine sheaves are soft, and soft sheaves are acyclic: \(H^i(X, \mathcal{F}) = 0\) for \(i > 0\). So a resolution of a sheaf by fine sheaves computes its cohomology.
On a smooth manifold \(M\), the sheaf \(\mathcal{C}^\infty_M\) of smooth functions is fine, using a smooth partition of unity subordinate to the cover, and so is every sheaf of \(\mathcal{C}^\infty_M\)-modules, such as the sheaves \(\mathcal{A}^p\) of smooth \(p\)-forms. The de Rham complex \(0 \to \ul{\RR} \to \mathcal{A}^0 \to \mathcal{A}^1 \to \cdots\) is a fine resolution of the constant sheaf, which proves \(H^p(M, \RR) \cong H^p_{\mathrm{dR}}(M)\). The sheaf of holomorphic functions on \(\CC\) is not soft: a function holomorphic near the closed set \(\{0, 1\}\), equal to \(0\) near \(0\) and \(1\) near \(1\), does not extend to an entire function.