Exact sequence of sheaves and associated sequence of graded modules Let $(X,\mathcal{O}_X)$ with $X=\mathbb{P}^n$ and consider a exact sequence of sheaves of $\mathcal{O}_X$-modules
$$0 \to \mathcal{F} \to \mathcal{H} \to \mathcal{G} \to 0 $$
Suppose that we apply the functor $\Gamma^*(\mathcal{F})=\bigoplus_{m \in \mathbb{Z}}\Gamma(X,\mathcal{F}(n))$ to the sequence (the functor in general is only left exact) and it remains exact and splits.
What can we say about the original sequence?  Does the sequence split?
 A: I will give an outline of the proof of this fact if all the sheaves involved are quasi-coherent. I claim that the original exact sequence does split.
First recall the Proj construction. Let $A=\oplus_{n\geq 0}A_n$ be a graded commutative ring, then $\operatorname{Proj} A$ is as a set the set of homogeneous prime ideas of $A$ not containing the trivial ideal $A_+=\oplus_{n>0}A_n$. For any homogeneous $f\in A$, $\operatorname{Proj}A$ has distinguished affine open $\operatorname{Spec}(A_f)_0$ (where $A_f$ is the localization by $f$ and $(A_f)_0$ means take the degree zero part of that). For a graded $A$ module $M$, we associate a quasicoherent sheaf $\tilde{M}$ by having its value on $(A_f)_0$ be $(M_f)_0$.
First I claim for for any quasi-coherent sheaf $\mathscr{M}$ on $\mathbb{P}^n$, $\widetilde{\Gamma^*(\mathscr{M})}$ is canonically isomorphic to $\mathscr{M}$. To show this you must use that the graded ring defining $\mathbb{P}^n$ is finitely generated and generated in degree $1$. 
Let $A$ still be a graded ring as above. Then if $N$ and $M$ are graded $A$ modules, and $N\to M$ a morphism, it induces a map $\tilde{N}\to \tilde{M}$. One shows that if $N\to M$ is injective (resp surjective, bijective) in degrees $n\gg 0$, then $\tilde{N}\to \tilde{M}$ is an injection (resp surjection, isomorphism).
Together the last two paragraphs are enough to imply your desired statements, as the isomorphisms $\widetilde{\Gamma^*(\mathscr{M})}\cong \mathscr{M}$ get you a splitting map $\mathcal{G}\to \mathcal{H}$ from that on the level of $\Gamma^*$. Then by the last paragraph, this identifies $\mathcal{G}$ with $\mathcal{H}\oplus \mathcal{F}$ as it does on the level of $\Gamma^*$.
