Let $\mathcal F$ be a sheaf on a topological space $X$ and $\mathcal U=\{U_i\}$ an open cover of $X$ If $\mathcal F$ is acyclic on every finite intersection of elements of $\mathcal U$ then $$ \check H^q(\mathcal U,\mathcal F)=\check H^q(X,\mathcal F), $$ where $\check H^q(\mathcal U,\mathcal F)$ is the $q$ th Cech cohomology group of $\mathcal F$ with respect to the open cover $\mathcal U$

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Bonavero, Laurent. Cohomology of Line Bundles on Toric Varieties, Vanishing Theorems. Lectures 16-17 from ``Summer School 2000: Geometry of Toric Varieties.''