relation between sheaf of hom and hom of sheaf

Question

If $\mathcal{M，N}$ are the associated sheaf of $A$ modules $M$ and $N$ on $X=Spec A,$ then what is $\mathcal{Hom_{O_X}(M,N)}$？Is that the associated sheaf of $Hom(M,N)\ ?$

Answer 1

Just to give a few more details on Daniel's comments:

In general, $\mathcal{H}om_{\mathcal{O}_X}(\mathcal{M},\mathcal{N})$ is not the associated sheaf to $Hom_A(M,N)$. Simple example: Take $A = \mathbb{Z}$, $M = \mathbb{Z}[\frac12]$ and $N = \mathbb{Z}$. Then $Hom_A(M,N) = 0$, but the Hom-sheaf is non-zero evaluated at the non-vanishing locus of $2$ (also known as $Spec \mathbb{Z}[\frac12]$).

Now assume that $M$ is a finitely presented module. Since taking associated module is left adjoint to global sections, we have a canonical map $$\widetilde{Hom_A(M,N)} \to \mathcal{H}om_{\mathcal{O}_X}(\mathcal{M},\mathcal{N}).$$ It is enough to show that this is an isomorphism on the standard opens $D(f)$ for $f \in A$. If we evaluate the source on $D(f)$, we get $Hom_A(M,N)[\frac 1f]$. If we evaluate the target on $D(f)$, we get $Hom_{A[\frac 1f]}(M[\frac1f], N[\frac1f]) \cong Hom_A(M, N[\frac1f])$. The localization $N[\frac1f]$ is isomorphic to the direct limit of

$$ N \xrightarrow{f\cdot} N \xrightarrow{f\cdot}\cdots. $$

Taking Hom out of a finitely presented module commutes with direct limits. Thus, $$Hom_A(M, N[\frac1f]) \cong Hom_A(M, N)[\frac1f]$$ as was to be shown.