Example 35.10.5. Let $S$ be a scheme. Let $\mathcal{F}$ and $\mathcal{G}$ be quasi-coherent modules on $(\mathit{Sch}/S)_\tau$ for one of the topologies $\tau$ considered in Lemma 35.10.4. In general it is not the case that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G})$ is quasi-coherent even if $\mathcal{F}$ is of finite presentation. Namely, say $S = \mathop{\mathrm{Spec}}(\mathbf{Z})$, $\mathcal{F} = \mathop{\mathrm{Coker}}(2 : \mathcal{O} \to \mathcal{O})$, and $\mathcal{G} = \mathcal{O}$. Then $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) = \mathcal{O}[2]$ is equal to the $2$-torsion in $\mathcal{O}$, which is not quasi-coherent.

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