Flasque sheaf

In mathematics, a flasque sheaf is a sheaf <math>\mathcal{F}<math> with the following property: if <math>X<math> is the base space and <math>U \subset V \subset X<math> are open subsets, then the restriction map <math>\Gamma(V, \mathcal{F}) \to \Gamma(U, \mathcal{F})<math> is surjective as a map of groups (rings, modules, etc.).

Flasque sheaves are useful because (by definition) sections of them extend. This means that they are some of the simplest sheaves to handle in terms of homological algebra. Flasque resolutions, that is, resolutions by means of flasque sheaves, are one approach to defining sheaf cohomology.

Flasque is a French word that has sometimes been translated into English as flabby.