= Flasque resolution
{title2=$0\to\mathcal F\to\mathcal I^0\to\mathcal I^1\to\cdots$}
A flasque resolution is an <exact sequence> of <sheaves of abelian groups> whose terms $\mathcal I^j$ are <flasque sheaves>. Every such sheaf embeds by its section germs into the sheaf $U\mapsto\prod_{P\in U}\mathcal F_P$. Its restriction maps are projections, so it is flasque. Apply this construction successively to the cokernels to obtain a resolution. The <sheaf cohomology> of $\mathcal F$ is the cohomology of the <cochain complex> $\Gamma(X,\mathcal I^\bullet)$. A quotient of two flasque sheaves in a <short exact sequence of sheaves> is flasque: the flasque kernel allows local lifts to be glued into a section of the middle sheaf over the smaller open set; extend that lift using flasqueness of the middle sheaf, and project. Consequently the resolution's successive cokernels are flasque when $\mathcal F$ is flasque, and exactness of sections shows that all its positive-degree cohomology vanishes.
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