I have been reading about local cohomology from Hartshorne's notes on the same and I have the following question.
Let $X$ be a topological space, and let $Z\subset X$ be closed. Then we have a long exact sequence $$0\to \Gamma_Z(X,\mathscr{F})\to \Gamma(X,\mathscr{F})\to \Gamma(X\backslash Z,\mathscr{F})\to H^1_Z(X,\mathscr{F})\to H^1(X,\mathscr{F}) \to H^1(X\backslash Z,\mathscr{F})\to \ldots $$
It is clear that if $H^1_Z(X,\scr{F}) = 0$, then every $s\in \Gamma(X\backslash Z, \scr{F})$ is the restriction of some element of $\Gamma(X,\scr{F})$. Moreover, if $H^1(X,\scr{F})$ somehow vanishes (for example, if we consider affine schemes $X$ and quasicoherent sheaves $\scr{F}$ on them), then $\Gamma(X,\mathscr{F})\to \Gamma(X\backslash Z, \scr{F})$ being surjective implies that $H^1_Z(X,\scr{F}) = 0$.
My question is, if $\Gamma(X,\mathscr{F})\to \Gamma(X\backslash Z,\scr{F})$ is surjective, then can we conclude in general that $H^1_Z(X,\scr{F})$ is zero? I remember hearing something along these lines holds, but I since I have not been able to find a proof of this, I'm not sure if this is true.