 Godement resolution

Embed a sheaf of abelian groups $\mathcal{F}$ into the sheaf $C(\mathcal{F})(U)={\prod }_{P\in U}{\mathcal{F}}_P$ by all its germs. Restrictions are projections, so $C(\mathcal{F})$ is flasque. Repeat on the successive sheaf cokernels to obtain a canonical flasque resolution. Its complex of global sections computes sheaf cohomology. When $\mathcal{F}$ is itself flasque, the flasque-kernel section-lifting lemma makes every successive quotient flasque and makes the global-section complex exact in positive degrees, proving $H^q(X,\mathcal{F})=0$ for $q>0$ without assuming that vanishing in the construction.

Godement resolution is a mathematical construct used in the field of algebraic geometry and homological algebra. It refers to a particular type of resolution of a sheaf (or an algebraic object) that provides insight into its structure via complex of sheaves or modules. More specifically, the Godement resolution is an injective resolution of a sheaf on a topological space, particularly within the context of sheaf theory. It is named after the mathematician Rémy Godement.