Lemma 17.22.8. Let $X$ be a ringed space. Let $I$ be a preordered set and let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system over $I$ consisting of sheaves of $\mathcal{O}_ X$-modules (see Categories, Section 4.21). Assume

1. $I$ is directed,

2. $\mathcal{G}$ is an $\mathcal{O}_ X$-module of finite presentation, and

3. $X$ has a cofinal system of open coverings $\mathcal{U} : X = \bigcup _{j\in J} U_ j$ with $J$ finite and $U_ j \cap U_{j'}$ quasi-compact for all $j, j' \in J$.

Then we have

$\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}_ i) = \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i).$

Proof. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathop{\mathrm{colim}}\nolimits \mathcal{F}_ i)$ and $\mathcal{H}_ i = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F}_ i)$. Recall that

$\mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}) = \Gamma (X, \mathcal{H}) \quad \text{and}\quad \mathop{\mathrm{Hom}}\nolimits _ X(\mathcal{G}, \mathcal{F}_ i) = \Gamma (X, \mathcal{H}_ i)$

by construction. By Lemma 17.22.7 we have $\mathcal{H} = \mathop{\mathrm{colim}}\nolimits \mathcal{H}_ i$. Thus the lemma follows from Sheaves, Lemma 6.29.1. $\square$

Comment #4323 by Minh-Tien Tran on

I think the proof only show that the map $\text{colim}_i \text{Hom}_{X} (\mathcal{G},\mathcal{F}_i)\to \text{Hom}_{X} (\mathcal{G},\text{colim}_i \mathcal{F}_i)$ is surjective. It is not very clear to me why this map is also an injection.

Comment #4479 by on

THanks for pointing this out. This was a bit of a brain teaser. We never use the lemma in full strenghth (namely we use it only when the $U_j$ can be chosen quasi-compact as well), but since we claimed it was true we should prove the whole thing as well... The fix is here.

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