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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GQN. Beware of the difference between the letter 'O' and the digit '0'.