Can anyone give an example of a sheaf that is supple, but not flabby?
Consider sheafs $\mathcal{F}$ of Abelian groups over $X$.
- it is flabby if for any $U$ open subset of $X$, the restriction $\rho_{V,U} : \mathcal{F}(U)\to\mathcal{F}(V)$ is surjective.
- It is said supple if, for $Z_1,Z_2$ closed in $U$ and $Z=Z_1\cup Z_2$, then any $f \in \Gamma_Z(U,\mathcal{F})$ (a section with support contained in $Z$) can be split as a sum $f_1+f_2$, with $f_i\in\Gamma_{Z_i}(U,\mathcal{F})$.
A flabby sheaf is supple, and the converse is not true, but I don't know any example... Actually I don't even know if supple is a standard notion or nomenclature.
Also, a bonus question. Take $Z_1\ldots Z_m$ closed subsets of $U$, and $Z=Z_1\cup Z_m$. Take a given sequence $f_i \in \Gamma_{Z_i}(U,\mathcal{F})$ such that $\sum f_i=0$. Then if $\mathcal{F}$ is flabby, we can split antissymetrically each $f_i$ as $f_i = \sum f_{ij}$ with $f_{ij} = -f_{ji} \in\Gamma_{Z_i\cap Z_j}(U,\mathcal{F})$. Is the same possible for supple sheafs?