Flasque or Flabby Sheaves
A flasque sheaf (also called a flabby sheaf) is a sheaf with the following property: if is the base topological space on which the sheaf is defined and
are open subsets, then the restriction map
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. Any sheaf has a canonical embedding into the flasque sheaf of all possibly discontinuous sections of the étalé space, and by repeating this we can find a canonical flasque resolution for any sheaf. 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.
Flasque sheaves are soft and acyclic.
Read more about this topic: Injective Sheaf
Famous quotes containing the words flabby and/or sheaves:
“His moving impulse is no flabby yearning to teach, to expound, to make simple; it is that “obscure inner necessity” of which Conrad tells us, the irresistible creative passion of a genuine artist, standing spell-bound before the impenetrable enigma that is life, enamoured by the strange beauty that plays over its sordidness, challenged to a wondering and half-terrified sort of representation of what passes understanding.”
—H.L. (Henry Lewis)
“A thousand golden sheaves were lying there,
Shining and still, but not for long to stay—
As if a thousand girls with golden hair
Might rise from where they slept and go away.”
—Edwin Arlington Robinson (1869–1935)