Lemma 56.7.1. Let $X$ and $Y$ be Noetherian schemes. Let $F : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ Y)$ be a functor. Then $F$ extends uniquely to a functor $\mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ which commutes with filtered colimits. If $F$ is additive, then its extension commutes with arbitrary direct sums. If $F$ is exact, left exact, or right exact, so is its extension.
Proof. The existence and uniqueness of the extension is a general fact, see Categories, Lemma 4.26.2. To see that the lemma applies observe that coherent modules are of finite presentation (Modules, Lemma 17.12.2) and hence categorically compact objects of $\textit{Mod}(\mathcal{O}_ X)$ by Modules, Lemma 17.22.8. Finally, every quasi-coherent module is a filtered colimit of coherent ones for example by Properties, Lemma 28.22.3.
Assume $F$ is additive. If $\mathcal{F} = \bigoplus _{j \in J} \mathcal{H}_ j$ with $\mathcal{H}_ j$ quasi-coherent, then $\mathcal{F} = \mathop{\mathrm{colim}}\nolimits _{J' \subset J\text{ finite}} \bigoplus _{j \in J'} \mathcal{H}_ j$. Denoting the extension of $F$ also by $F$ we obtain
\begin{align*} F(\mathcal{F}) & = \mathop{\mathrm{colim}}\nolimits _{J' \subset J\text{ finite}} F(\bigoplus \nolimits _{j \in J'} \mathcal{H}_ j) \\ & = \mathop{\mathrm{colim}}\nolimits _{J' \subset J\text{ finite}} \bigoplus \nolimits _{j \in J'} F(\mathcal{H}_ j) \\ & = \bigoplus \nolimits _{j \in J} F(\mathcal{H}_ j) \end{align*}
Thus $F$ commutes with arbitrary direct sums.
Suppose $0 \to \mathcal{F} \to \mathcal{F}' \to \mathcal{F}'' \to 0$ is a short exact sequence of quasi-coherent $\mathcal{O}_ X$-modules. Then we write $\mathcal{F}' = \bigcup \mathcal{F}'_ i$ as the union of its coherent submodules, see Properties, Lemma 28.22.3. Denote $\mathcal{F}''_ i \subset \mathcal{F}''$ the image of $\mathcal{F}'_ i$ and denote $\mathcal{F}_ i = \mathcal{F} \cap \mathcal{F}'_ i = \mathop{\mathrm{Ker}}(\mathcal{F}'_ i \to \mathcal{F}''_ i)$. Then it is clear that $\mathcal{F} = \bigcup \mathcal{F}_ i$ and $\mathcal{F}'' = \bigcup \mathcal{F}''_ i$ and that we have short exact sequences
\[ 0 \to \mathcal{F}_ i \to \mathcal{F}_ i' \to \mathcal{F}_ i'' \to 0 \]
Since the extension commutes with filtered colimits we have $F(\mathcal{F}) = \mathop{\mathrm{colim}}\nolimits _{i \in I} F(\mathcal{F}_ i)$, $F(\mathcal{F}') = \mathop{\mathrm{colim}}\nolimits _{i \in I} F(\mathcal{F}'_ i)$, and $F(\mathcal{F}'') = \mathop{\mathrm{colim}}\nolimits _{i \in I} F(\mathcal{F}''_ i)$. Since filtered colimits are exact (Modules, Lemma 17.3.2) we conclude that exactness properties of $F$ are inherited by its extension. $\square$
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