(Godement resolution)
Let \(\mcf\) be a sheaf of abelian groups on a topological space \(X\). Put \[ \mathcal{G}^0(\mcf)(U) = \prod_{x \in U} \mcf_x , \] the sheaf of all, not necessarily continuous, sections of the espace étalé, with the injection \(\mcf \to \mathcal{G}^0(\mcf)\) sending \(s\) to its germs. Inductively, with \(\mathcal{Z}^0 = \mcf\) and \(\mathcal{Z}^{n+1} = \operatorname{coker}\big(\mathcal{Z}^{n} \to \mathcal{G}^0(\mathcal{Z}^{n})\big)\), set \(\mathcal{G}^n(\mcf) = \mathcal{G}^0(\mathcal{Z}^n)\) with differential \(\mathcal{G}^n(\mcf) \to \mathcal{Z}^{n+1} \to \mathcal{G}^{n+1}(\mcf)\). The complex \(0 \to \mcf \to \mathcal{G}^0(\mcf) \to \mathcal{G}^1(\mcf) \to \cdots\) is the Godement resolution.
Each \(\mathcal{G}^n(\mcf)\) is flasque, since a product of stalks over a smaller open set extends by zero to a larger one.
The Godement resolution is exact and functorial in \(\mcf\), and exact as a functor of \(\mcf\), because it is built from stalks.
Flasque sheaves are acyclic, so \(H^i(X, \mcf) = H^i\big(\Gamma(X, \mathcal{G}^\bullet(\mcf))\big)\).
For a continuous map \(f\colon X\to Y\), pushforwards of flasque sheaves are flasque and flasque sheaves are \(f_*\)-acyclic, so \(R^if_*\mcf\cong\mathcal H^i\big(f_*\mathcal G^\bullet(\mcf)\big)\). The complexes \(\Gamma(X,\mathcal G^\bullet(\mcf))\) and \(f_*\mathcal G^\bullet(\mcf)\) are functorial in \(\mcf\) and involve no choice of injective resolution.
Dependencies
D-COHFLQFlasque sheaves and acyclicityuses