Flabby sheaf

2020 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL]

A flabby sheaf" is a sheaf $F$ of sets over a topological space $X$ such that for any set $U$ open in $X$ the restriction mapping $F(X)\to F(U)$ is surjective. Examples of such sheaves include the sheaf of germs of all (not necessarily continuous) sections of a fibre space with base $X$, the sheaf of germs of divisors (cf. Divisor), and a prime sheaf $F$ over an irreducible algebraic variety. Flabbiness of a sheaf $F$ 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 $X$ is paracompact, a flabby sheaf is a soft sheaf, i.e. any section of $F$ over a closed set can be extended to the entire space $X$.

Let

$$0\to F^0\to F^1\to \cdots$$ be an exact sequence of flabby sheaves of Abelian groups. Then, for any family $\Phi$ of supports, the corresponding sequence of sections (the supports of which belong to $\Phi$)

$$0\to\def\G{\Gamma}\G_\Phi(F^0)\to\G_\Phi(F^1)\to\cdots$$ is exact, i.e. $F\mapsto \G_\Phi(F)$ 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, [Mi].

References

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