Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 118 / 2 / a / Solution

Choose a flasque resolution of the sheaf of abelian groups $\mathcal{F}$,A flasque sheaf has surjective restriction maps; such resolutions exist for every sheaf of abelian groups. Apply the global section functor to obtain a cochain complex. The sheaf cohomology groups areFor $i=0$, the denominator is zero and $H^0(X,\mathcal{F})=\Gamma (X,\mathcal{F})$. Different resolutions give canonically isomorphic groups; the resolution principle for sheaf cohomology is what permits other acyclic resolutions to compute these same groups.