Hom sheaf over a scheme in the case of quasi-coherent sheaf at first argument

Let $X$ be a scheme and $\mathcal{F},\mathcal{G}$ be two sheaves of $\mathcal{O}_X$-modules. I showed that the presheaf which assigns each open subset $U$ of $X$, $$U \longmapsto \mathrm{Hom}_{\mathcal{O}_X}(\mathcal{F}|_U,\mathcal{G}|_U)$$ is a sheaf. Let us denote it by $\mathcal{H} = Hom_{\mathcal{O}_X}(\mathcal{F},\mathcal{G})$.

In [Liu, "Algebraic Geometry and Arithmetic Curves"], page 172, Ex 1.5 we need to prove the following assertion (and I quote):

Let $X = \operatorname{Spec}(A)$ be an affine scheme and $\mathcal{F}$ a quasi-coherent $\mathcal{O}_X$-module (as a sheaf). Then he canonical map $$\mathrm{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G}) \longrightarrow \mathrm{Hom}_A (\mathcal{F}(X),\mathcal{G}(X))$$ is a bijection.

In the LHS we have a sheaf, in the RHS with an $A$-module. I suspect there is a mistake in the question and the author ment $$\mathcal{H}(X) = \left( \mathrm{Hom}_{\mathcal{O}_X}(\mathcal{F},\mathcal{G}) \right)(X) \longrightarrow \mathrm{Hom}_A (\mathcal{F}(X),\mathcal{G}(X))$$ which makes more sense.

To prove this or the corrected statement I need to use two facts:

1. Affine scheme may be covered by principal open subsets $D(f) = \{ x \in X \mid f \notin x \}$ where $x = \mathfrak{p} \in \operatorname{Spec}(A)=X$ is a prime ideal.
2. The $\mathcal{F}$ is quasi-coherent and thus for every prime ideal $x \in X = \operatorname{Spec}(A)$ the stalk $\mathcal{F}_x$ is the localization of $\mathcal{F}(X)$ at $x$, that is $(\mathcal{F}(X))_x$.
• No, the exercise is correct as written. There is a difference between $\mathcal{H}om$ and $\mathrm{Hom}$! – Zhen Lin May 28 '14 at 13:15
• The key to understand this is that an element of $\mathcal H = \mathcal{H}om_{\mathcal O_X}(\mathcal F,\mathcal G)$ is just a local homomorphism of sheaves, whereas an element of $\mathrm{Hom}_{\mathcal O_X}(\mathcal F, \mathcal G)$ is a homomorphism of sheaves. In particular, you need to give a homomorphism $\mathcal F(X) \to \mathcal G(X)$. – Fredrik Meyer May 28 '14 at 13:55
• @GeorgesElencwajg Except from one typoes all the other occurences said "quasi-coherent". – LinAlgMan May 29 '14 at 8:51
• Thank you for the clarifications. Now the question makes more sense. So far I managed to construct the canonical map which sends each $\varphi : \mathcal{F} \to \mathcal{G}$ to $\varphi(X) : \mathcal{F}(X) \to \mathcal{G}(X)$ and prove it is injective. I now try to prove it is surjective and for this I will probably need the fact that $X$ is affine scheme and $\mathcal{F}$ is quasi-coherent. I managed to prove it if I know also that $\mathcal{G}$ is quasi-coherent. In that case, I build the localized maps $\varphi(D(f_i))$ and glue them to $\varphi(U)$ and from that obtain $\varphi$. – LinAlgMan May 29 '14 at 9:38