A question on flasque sheaf Let $0\to \mathscr{F}'\to\mathscr{F}\to\mathscr{F}''\to 0$ be an exact sequene of sheaves. It is well known that $\mathscr{F}$ flasque iff $\mathscr{F}''$ flasque provided $\mathscr{F}'$ is flasque. How about $\mathscr{F}'$ if the other two sheaves are flasque? Can we prove $\mathscr{F}'$ flasque or end the question by a counterexample? Thanks for any tips.
 A: For a naturally-arising example, take $X$ to be a smooth variety and consider
$0 \to \mathcal{O}_X^\times \to \mathcal{R}_X^\times \to \mathcal{D}\textit{iv}_X \to 0$
where $\mathcal{O}_X$ is the structure sheaf, $\mathcal{R}_X$ is the sheaf of rational functions, and $\mathcal{D}\textit{iv}_X$ is (by definition) the sheaf of Cartier divisors.
A: A counterexample: $0\to \mathscr{O}_\mathbb Z\to \mathbb Q \to \mathbb Q/\mathscr{O}_\mathbb Z \to 0$ where $\mathscr{O}_\mathbb Z$ is the structure sheaf and $\mathbb Q$ is the constant sheaf on $Spec \mathbb Z$.
A: I think this is a counterexample: consider a topological space with three points, $X = \{u, v, p\}$, whose open sets are $X$, $\{u, v\}$, $\{u\}$, $\{v\}$ and $\emptyset$. (It is the topological space associated to the $V$-shaped poset on three elements.)
Note that $X$ is the only open subset containing $p$. Also, $U := \{u\}$, resp. $V := \{v\}$, is the smallest open subset containing $u$, resp. $v$.
It follows that for any abelian sheaf $\mathscr F$ we have
$$\mathscr F_p = \Gamma(X, \mathscr F),\qquad
\mathscr F_u = \Gamma(U, \mathscr F),\qquad
\mathscr F_v = \Gamma(V, \mathscr F).$$
Now let $\mathscr F' = \underline{\mathbb Z}_X$. It is not flabby (consider the restriction from $X$ to $\{u, v\}$). It injects canonically into $\underline{\mathbb Z}_u \oplus \underline{\mathbb Z}_v \oplus \underline{\mathbb Z}_p$. The cokernel is supported at $p$, hence flabby.
