I know this question is absolutely trivial, but having self-studied the subject I feel extremely unsure on the basics. Do not hesitate to downgrade the question, if you feel it deserves so.


Given a morphism of complex of sheaves $\varphi:\mathcal{F}^\bullet\to \mathcal{G}^\bullet$ on a topological space $X$ (eventually as nice as needed), we have two induced morphisms, one in local cohomology, $\mathcal{H}^\bullet(\varphi):\mathcal{H}^\bullet(\mathcal{F}^\bullet)\to \mathcal{H}^\bullet(\mathcal{G}^\bullet)$, and one in global cohomology $H^\bullet(\varphi):H^\bullet(X;\mathcal{F}^\bullet)\to H^\bullet(X,\mathcal{G}^\bullet)$.


The question is: does $\mathcal{H}^\bullet(\varphi)=0$ imply $H^\bullet(\varphi)=0$? 


I guess so, since there is a spectral sequence with $E\_2$-page given by cohomology with cefficients in local cohomology abutting to global cohomology. But as I said, I do not trust myself too much on self-study..



Does trivial on local cohomology implies trivial on global cohomology?
The exact sequence
$0\rightarrow\mathrm{Z}/p\rightarrow\mathrm{Z}/p^2\rightarrow\mathrm{Z}/p\rightarrow0$
($p$ a prime say) gives a map $\mathrm{Z}/p\rightarrow\mathrm{Z}/p[1]$ in the
derived category of abelian groups (which can be realised as a map of
complexes). It clearly induces zero on cohomology. We can then take any
topological space $X$ and pull this back to $X$ giving a map (in the derived
category of sheaves on $X$) between constant sheaves again inducing zero in
(local) cohomology. The maps $H^i(X,\mathrm{Z}/p)\rightarrow
H^{i+1}(X,\mathrm{Z}/p)$ induced by this map are the Bockstein operations. Hence
we can let $X$ be any space with non-zero Bockstein map. A particular case for
$p=2$ are the real projective spaces of dimension at least $2$.


