Sheaf cohomology as the derived functor of global sections

(Enough injectives)

The construction requires enough injective objects. For each \(x\), embed the stalk \(\mcf_x \injects I_x\) into an injective \(\OO_{X,x}\dash\)module, let \(j^x: \ts{x} \injects X\), and set \(\mci \da \prod_{x \in X} j^x_* I_x\). Then \(\Hom(\mcg, \mci) = \prod_x \Hom_{\OO_{X,x}}(\mcg_x, I_x)\), a composite of the exact stalk functors with exact \(\Hom\) functors, so \(\mci\) is injective, and \(\mcf \injects \mci\).

(Acyclic resolutions)

For every resolution \(\mcf\to\mcl^\bullet\) by \(\globsec{X;\wait}\dash\)acyclic sheaves, \(H^i(X,\mcf)\cong H^i(\globsec{X;\mcl^\bullet})\) [@Har10a, Proposition III.1.2A]. Flasque sheaves are acyclic [@Har10a, Proposition III.2.5], and every sheaf embeds in the flasque sheaf \(U\mapsto\prod_{x\in U}\mcf_x\) of discontinuous sections. Hence \(H^i\) is effaceable for \(i>0\), and \((H^i)_{i\ge0}\) is a universal \(\delta\dash\)functor [@Har10a, Theorem III.1.3A]: for every \(\delta\dash\)functor \((T^i)\) from \(\Sh(X;\Ab)\) to \(\Ab\), every natural transformation \(H^0\to T^0\) extends uniquely to a morphism of \(\delta\dash\)functors \((H^i)\to(T^i)\).