Acyclic resolution theorem ($H^q(X,\mathcal{F})\cong H^q(\Gamma (X,{\mathcal{I}}^{\bullet }))$)

If $0\to \mathcal{F}\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$ is an exact resolution of sheaves and each ${\mathcal{I}}^j$ has vanishing positive sheaf cohomology, then the cohomology of its global-section complex computes $H^q(X,\mathcal{F})$. Fine sheaves provide acyclic terms on a paracompact manifold, giving the analytic Dolbeault resolution of holomorphic differential forms.