Godement resolution

The Godement resolution of a sheaf is a construction in homological algebra which allows one to view global, cohomological information about the sheaf in terms of local information coming from its stalks. It is useful for computing sheaf cohomology. It was discovered by Roger Godement.

Given a topological space X (more generally, a topos X with enough points), and a sheaf F on X, the Godement construction for F gives a sheaf ${\displaystyle \operatorname {Gode} (F)}$ constructed as follows. For each point ${\displaystyle x\in X}$, let ${\displaystyle F_{x}}$ denote the stalk of F at x. Given an open set ${\displaystyle U\subset X}$, define

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