Flabby sheaf

A sheaf of sets over a topological space such that for any set open in the restriction mapping is surjective. Examples of such sheaves include the sheaf of germs of all (not necessarily continuous) sections of a fibre space with base , the sheaf of germs of divisors (cf. Divisor), and a prime sheaf over an irreducible algebraic variety. Flabbiness of a sheaf is a local property (i.e. a flabby sheaf induces a flabby sheaf on any open set). A quotient sheaf of a flabby sheaf by a flabby sheaf is itself a flabby sheaf. The image of a flabby sheaf under a continuous mapping is a flabby sheaf. If is paracompact, a flabby sheaf is a soft sheaf, i.e. any section of over a closed set can be extended to the entire space .

Let

be an exact sequence of flabby sheaves of Abelian groups. Then, for any family of supports, the corresponding sequence of sections (the supports of which belong to )

is exact, i.e. is a left-exact functor.

Comments

Flabby sheaves are used for resolutions in the construction of sheaf cohomology (i.e. cohomology with values in a sheaf) in algebraic geometry and topology, [a1].

References

[a1]	J.S. Milne, "Etale cohomology" , Princeton Univ. Press (1980) MR0559531 Zbl 0433.14012