The Stacks project

Lemma 18.27.11. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be an $\mathcal{O}$-module of finite presentation. Let $\mathcal{G} = \mathop{\mathrm{colim}}\nolimits _{\lambda \in \Lambda } \mathcal{G}_\lambda $ be a filtered colimit of $\mathcal{O}$-modules. Then the canonical map

\[ \mathop{\mathrm{colim}}\nolimits _\lambda \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}_\lambda ) \longrightarrow \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \]

is an isomorphism.

Proof. It suffices to show the arrow is an isomorphism after restriction to $U$ for all $U$ in $\mathcal{C}$. Both taking colimits of sheaves of modules and taking internal hom commute with restriction to $U$. See for example Lemmas 18.14.3 and 18.27.2. Fix $U$. Given a covering $\{ U_ i \to U\} _{i \in I}$, then it suffices to prove the restriction to each $U_ i$ is an isomorphism. Hence we may assume $\mathcal{F}$ has a global presentation

\[ \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{O} \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{O} \to \mathcal{F} \to 0 \]

The functor $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(-, -)$ commutes with finite direct sums in either variable and $\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{O}, -)$ is the identity functor. By this and by Lemma 18.27.5 we obtain an exact sequence

\[ 0 \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \to \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G} \to \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G} \]

Since filtered colimits are exact in $\textit{Mod}(\mathcal{O})$ by Lemma 18.14.2 also the top row in the following commutative diagram is exact

\[ \xymatrix{ 0 \ar[r] & \mathop{\mathrm{colim}}\nolimits _\lambda \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}_\lambda ) \ar[r] \ar[d] & \mathop{\mathrm{colim}}\nolimits _\lambda \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G}_\lambda \ar[r] \ar[d] & \mathop{\mathrm{colim}}\nolimits _\lambda \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G}_\lambda \ar[d] \\ 0 \ar[r] & \mathop{\mathcal{H}\! \mathit{om}}\nolimits _\mathcal {O}(\mathcal{F}, \mathcal{G}) \ar[r] & \bigoplus \nolimits _{i = 1, \ldots , n} \mathcal{G} \ar[r] & \bigoplus \nolimits _{j = 1, \ldots , m} \mathcal{G} } \]

Since the right two vertical arrows are isomorphisms we conclude. $\square$

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