Injective resolution of sheaves ($0\to \mathcal{F}\to {\mathcal{J}}^0\to {\mathcal{J}}^1\to \cdots$)

An injective resolution is an exact sequence starting with a sheaf and continuing with injective sheaves. It is constructed by successively embedding each quotient in an injective sheaf. Taking global sections and then cohomology computes sheaf cohomology; in particular an injective sheaf has zero higher cohomology.