Godement resolution of an abelian sheaf

Let $X$ be a topological space and let $\mathcal{F}$ be a sheaf of abelian groups on $X$ (A sheaf on a topological space). For $x\in X$ let ${\mathcal{F}}_x$ be the stalk at $x$ (The stalk of a presheaf at a point, Germs of sections) and let $i_{x,\ast }A$ be the skyscraper sheaf at $x$ with value an abelian group $A$ (A skyscraper sheaf of abelian groups at a point): $(i_{x,\ast }A)(V)=A$ when $x\in V$ and $(i_{x,\ast }A)(V)=0$ otherwise.

Put $C^0(\mathcal{F}):={\prod }_{x\in X}{i}_{x,\ast }{\mathcal{F}}_x,$ the product in $\mathrm{A}\mathrm{b}(X)$ of the skyscraper sheaves of the stalks of $\mathcal{F}$; since products of sheaves are computed open by open, for an open $V\subseteq X$ one has $C^0(\mathcal{F})(V)={\prod }_{x\in V}{\mathcal{F}}_x\mathrm{.}$ The germ maps $\mathcal{F}(V)\to {\prod }_{x\in V}{\mathcal{F}}_x$, $s\mapsto (s_x{)}_{x\in V}$ (Germs of sections) are compatible with restrictions and assemble into a morphism of sheaves ${\epsilon }_{\mathcal{F}}:\mathcal{F}\to C^0(\mathcal{F})$, the germ map of $\mathcal{F}$. Let $Q^0(\mathcal{F}):=\mathrm{coker}({\epsilon }_{\mathcal{F}})$ be the cokernel sheaf of ${\epsilon }_{\mathcal{F}}$ (Kernel sheaves are objectwise, while cokernels and images are sheafified). Recursively, having defined a sheaf $Q^n(\mathcal{F})$ of abelian groups on $X$, put $C^{n+1}(\mathcal{F}):=C^0(Q^n(\mathcal{F})),Q^{n+1}(\mathcal{F}):=\mathrm{coker}(Q^n(\mathcal{F})\stackrel{{\epsilon }_{Q^n}}{\to }{C}^{n+1}(\mathcal{F})),$ with ${\epsilon }_{Q^n}$ the germ map of $Q^n(\mathcal{F})$. The Godement resolution of $\mathcal{F}$ is the coaugmented complex $0\to \mathcal{F}\stackrel{{\epsilon }_{\mathcal{F}}}{\to }{C}^0(\mathcal{F})\stackrel{d^0}{\to }{C}^1(\mathcal{F})\stackrel{d^1}{\to }\cdots ,$ where $d^n:C^n(\mathcal{F})\to C^{n+1}(\mathcal{F})$ is the composite of the quotient morphism $C^n(\mathcal{F})\to Q^n(\mathcal{F})$ with the germ map ${\epsilon }_{Q^n}:Q^n(\mathcal{F})\to C^0(Q^n(\mathcal{F}))=C^{n+1}(\mathcal{F})$. Its terms are sheaves of abelian groups on $X$ and its differentials are differentials of the cochain complex $\mathrm{\Gamma }(X,C^{\bullet }(\mathcal{F}))$ after applying global sections.

The construction is functorial: a morphism $\phi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves induces morphisms of stalks ${\phi }_x$, hence morphisms $C^0(\phi )$, $Q^0(\phi )$, and by recursion morphisms $C^n(\phi )$ and $Q^n(\phi )$ commuting with the germ maps and the differentials, so that $C^{\bullet }$ is a functor from $\mathrm{A}\mathrm{b}(X)$ to the category of coaugmented cochain complexes in $\mathrm{A}\mathrm{b}(X)$ (Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories). The germ map ${\epsilon }_{\mathcal{E}}:\mathcal{E}\to C^0(\mathcal{E})$ is injective for every abelian sheaf $\mathcal{E}$, by A section of a sheaf of groups is zero exactly when all of its germs are zero.