Lemma 18.27.12. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } \mathcal{G}_\lambda $ be a filtered colimit of $\mathcal{O}$-modules. Let $\mathcal{F}$ be an $\mathcal{O}$-module of finite presentation. Then we have
\[ \mathop{\mathrm{colim}}\nolimits _\lambda \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}_\lambda ) = \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}). \]
if the hypotheses of Sites, Lemma 7.17.8 part (4) are satisfied for the site $\mathcal{C}$; please see Sites, Remark 7.17.9.
Proof.
Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathop{\mathrm{colim}}\nolimits \mathcal{G}_\lambda )$ and $\mathcal{H}_\lambda = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}_\lambda )$. Recall that
\[ \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) = \Gamma (\mathcal{C}, \mathcal{H}) \quad \text{and}\quad \mathop{\mathrm{Hom}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}_\lambda ) = \Gamma (\mathcal{C}, \mathcal{H}_\lambda ) \]
by construction. By Lemma 18.27.11 we have $\mathcal{H} = \mathop{\mathrm{colim}}\nolimits \mathcal{H}_\lambda $. Thus the lemma follows from Sites, Lemma 7.17.8.
$\square$
Comments (0)