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
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