= Solution
Choose a <flasque resolution> of the <sheaf of abelian groups> $\mathcal F$,
$$
0\longrightarrow\mathcal F\longrightarrow\mathcal I^0\xrightarrow{d^0}\mathcal I^1\xrightarrow{d^1}\cdots.
$$
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 are
$$
\boxed{H^i(X,\mathcal F)=
\frac{\ker\bigl(\Gamma(X,\mathcal I^i)\to\Gamma(X,\mathcal I^{i+1})\bigr)}
{\operatorname{im}\bigl(\Gamma(X,\mathcal I^{i-1})\to\Gamma(X,\mathcal I^i)\bigr)}.}
$$
For $i=0$, 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.
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