# The difference between the flabby sheaf and the fine sheaf

As we know,the flabby sheaf and the fine sheaf are all acyclic,but does one imply another?I need some examples to distinguish them,who can help me?

-

## 1 Answer

Sheaf of $C^{\infty}$ functions on paracompact manifold $X$ is fine because exists partitions of unity on $X$. But is not flabby because you cannot extend function on open subset $U$ if function blow up at frontier of $U$.
Constant sheaf with values in $\mathbb Z$ is flabby on irreducible algebraic variety but not fine.

-
Thanks for your help. –  Strongart Mar 22 '11 at 10:23