This is something I got stuck thinking about while trying to solve a problem in Hartshorne: if $\mathcal{F}$ is a quasi coherent sheaf on a noetherian scheme $X$ such that $\mathcal{F}|_U$ is flasque for all open affine $U\subset X$ will $\mathcal{F}$ then be flasque?
The problem I'm working on is III.3.6 (b) which says that any injective object of $QCoh(X)$ is flasque. I was able to prove the hint which says that for any injective sheaf $\mathcal{J}$ on $X$ and any open $U\subset X$ the restriction $\mathcal{J}|_U$ is an injective sheaf on $U$.
Best Answer
Just to expand my comment that being flasque is local. If $\mathcal F$ is a locally flasque sheaf over (a topological space) $X$, let $U\subset X$ be an open subset, and $a\in\mathcal F(U)$. We want to show that $a$ can be extended to be a glocal section over $X$. By Zorn's lemma, we may pick a maximal extension $\tilde{a}$ of $a$. If the domain $V$ of $a$ is not $X$, pick $x_0\in X\setminus V$, then by $\mathcal F$ is locally flasque, pick an open Flasque neighborhood $W$ of $x_0$, $\tilde{a}_{W\cap V}$ can be extended to a section $\tilde{a}'\in\mathcal F(W)$, but now glue $\tilde{a}$ and $\tilde{a}'$, we get a strict extension of $\tilde{a}$.