# 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 |

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Flabby sheaf.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Flabby_sheaf&oldid=33620