If f is flat, \mathcal{F}, \mathcal{G} \in \mathop{\mathrm{Ob}}\nolimits (\mathit{QCoh}(\mathcal{O}_\mathcal {X})), and \mathcal{F} of finite presentation and let then we have
d(hom(\mathcal{F}, \mathcal{G})) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(d(\mathcal{F}), d(\mathcal{G}))with notation as in Lemma 103.10.8. Perhaps the easiest way to see this is as follows
\begin{align*} d(hom(\mathcal{F}, \mathcal{G})) & = d(Q(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G}))) \\ & = c(\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})) \\ & = f^*\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {X}}(\mathcal{F}, \mathcal{G})|_{U_{\acute{e}tale}} \\ & = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_\mathcal {U}}(f^*\mathcal{F}, f^*\mathcal{G})|_{U_{\acute{e}tale}} \\ & = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ U}(f^*\mathcal{F}|_{U_{\acute{e}tale}}, f^*\mathcal{G}|_{U_{\acute{e}tale}}) \end{align*}The first equality by construction of hom. The second equality by (7). The third equality by definition of c. The fourth equality by Modules on Sites, Lemma 18.31.4. The final equality by the same reference applied to the flat morphism of ringed topoi i_ U (U_{\acute{e}tale}, \mathcal{O}_ U) \to (\mathcal{U}_{\acute{e}tale}, \mathcal{O}_\mathcal {U}) of Sheaves on Stacks, Lemma 96.10.1.
Comments (0)