Lemma 17.22.7. Let $X$ be a ringed space. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module of finite presentation. Let $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } \mathcal{G}_\lambda $ be a filtered colimit of $\mathcal{O}_ X$-modules. Then the canonical map
\[ \mathop{\mathrm{colim}}\nolimits _\lambda \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}_\lambda ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \]
is an isomorphism.
Proof.
Taking colimits of sheaves of modules commutes with restriction to opens, see Sheaves, Section 6.29. Hence we may assume $\mathcal{F}$ has a global presentation
\[ \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O}_ X \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O}_ X \to \mathcal{F} \to 0 \]
The functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(-, -)$ commutes with finite direct sums in either variable and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{O}_ X, -)$ is the identity functor. By this and by Lemma 17.22.2 we obtain an exact sequence
\[ 0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \to \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G} \to \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G} \]
Since filtered colimits are exact in $\textit{Mod}(\mathcal{O}_ X)$ also the top row in the following commutative diagram is exact
\[ \xymatrix{ 0 \ar[r] & \mathop{\mathrm{colim}}\nolimits _\lambda \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}_\lambda ) \ar[r] \ar[d] & \mathop{\mathrm{colim}}\nolimits _\lambda \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G}_\lambda \ar[r] \ar[d] & \mathop{\mathrm{colim}}\nolimits _\lambda \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G}_\lambda \ar[d] \\ 0 \ar[r] & \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G}) \ar[r] & \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G} \ar[r] & \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G} } \]
Since the right two vertical arrows are isomorphisms we conclude.
$\square$
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