Stalks of the sheaf $\mathscr{H}om$?

Question

The question is basically the title. What are the stalks of the sheaf $\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})$? If $X$ is a noetherian scheme and $\mathscr{F}$ is coherent, then $$\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})_p \cong \operatorname{Hom}_{\mathscr{O}_{X,p}}(\mathscr{F}_p,\mathscr{G}_p).$$ This is Proposition III.6.8 of Hartshorne. Is this true for more general $X$ and $\mathscr{F}$? Hartshorne remarks, "Note that even the case $i=0$ is not true without some special hypothesis on $\mathscr{F}$, such as $\mathscr{F}$ coherent." What counterexample does he have in mind? I suppose we use that $\mathscr{F}$ coherent to get a locally free resolution; of course, this doesn't mean that it's not true without this assumption.

Answer 1

Hartshorne often assumes schemes to be noetherian so that his definition of "coherence" is correct. The general and correct definition of coherence is a bit complicated; what Hartshorne defines as coherence is actually called "locally of finite type", which agrees with "locally of finite presentation" over noetherian schemes. It turns out that, over arbitrary (quasi-compact quasi-separated) schemes, the quasi-coherent sheaves locally of finite presentation behave very well and provide a perfect substitute for coherent sheaves over noetherian schemes.

If $X$ is any ringed space, and $F,G$ are $\mathcal{O}_X$-modules, and $p \in X$, there is a canonical homomorphism $$\underline{\hom}(F,G)_p \to \hom_{\mathcal{O}_{X,p}}(F_p,G_p).$$ We ask ourselves if this is an isomorphism. Let $C$ be the class of all "good" $F$ i.e. for which this is an isomorphism for all $G$. We have the following basic properties:

1) If there is an open neighborhood $U$ of $p$ such that $F|_U \in C$ (for $U$ instead of $X$), then $F \in C$.

2) $\mathcal{O}_X \in F$.

3) $C$ is closed under finite direct sums.

4) If $F \to F' \to F'' \to 0$ is a right exact sequence and $F,F' \in C$, then $F'' \in C$.

From these properties it follows formally that $C$ contains all sheaves of modules $F$ which are locally of finite presentation (around $p$).

The next case would be to consider $F=\mathcal{O}_X^{\oplus I}$, an arbitrary free $\mathcal{O}_X$-module. Then the homomorphism becomes the canonical map $$(\prod_{i \in I} G)_p \to \prod_{i \in I} G_p,~[(s_i)_i]_p \mapsto ([s_i]_p)_i$$ If $I$ is infinite, this is usually no isomorphism, even if $G=\mathcal{O}_X$. See math.SE/164960 for a counterexample. There we have $X=\mathbb{R}$, but there is a similar counterexample for schemes $X$.

Answer 2

An example outside the world of schmes is the following: Take $X=\mathbb{R}$ and $\mathcal{O}_X$ the constant sheaf $\mathbb{Z}$, and for $\mathscr{F}$ the subsheaf of $\mathcal{O}_X$ such that $\mathscr{F}_0=0$ and $\mathscr{F}_x=\mathbb{Z}$ for $x\neq 0$. Note that $\mathscr{F}|U$ is not the zero sheaf for any neighborhood of $0$ so the germ at 0 of the identity $\mathscr{F}\rightarrow\mathscr{F}$ is not zero. On the other side $Hom_\mathbb{Z}(\mathscr{F}_0,\mathscr{F}_0)=0$.