The Stacks project

Lemma 102.10.8. Let $\mathcal{X}$ be an algebraic stack. Let $\mathcal{F}$ be an $\mathcal{O}_\mathcal {X}$-module of finite presentation and let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_\mathcal {X}$-module. The internal homs $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ computed in $\textit{Mod}(\mathcal{X}_{\acute{e}tale}, \mathcal{O}_\mathcal {X})$ or $\textit{Mod}(\mathcal{O}_\mathcal {X})$ agree and the common value is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$. The quasi-coherent module $ hom(\mathcal{F}, \mathcal{G}) = Q(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})) $ has the following universal property

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H}, hom(\mathcal{F}, \mathcal{G})) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H} \otimes _{\mathcal{O}_\mathcal {X}} \mathcal{F}, \mathcal{G}) \]

for $\mathcal{H}$ in $\mathit{QCoh}(\mathcal{O}_\mathcal {X})$.

Proof. The construction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ in Modules on Sites, Section 18.27 depends only on $\mathcal{F}$ and $\mathcal{G}$ as presheaves of modules; the output $\mathop{\mathcal{H}\! \mathit{om}}\nolimits $ is a sheaf for the fppf topology because $\mathcal{F}$ and $\mathcal{G}$ are assumed sheaves in the fppf topology, see Modules on Sites, Lemma 18.27.1. By Sheaves on Stacks, Lemma 95.12.4 we see that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is locally quasi-coherent. By Lemma 102.7.2 we see that $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ has the flat base change property. Hence $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})$ is an object of $\textit{LQCoh}^{fbc}(\mathcal{O}_\mathcal {X})$ and it makes sense to apply the functor $Q$ of Lemma 102.10.1 to it. By the universal property of $Q$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H}, Q(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G}))) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {X}(\mathcal{H}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})) \]

for $\mathcal{H}$ quasi-coherent, hence the displayed formula of the lemma follows from Modules on Sites, Lemma 18.27.6. $\square$


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