Lemma 17.11.6. 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. An element of the left hand side is given by the equivalence classe of a pair $(i, \alpha _ i)$ where $i \in I$ and $\alpha _ i : \mathcal{G} \to \mathcal{F}_ i$ is a morphism of $\mathcal{O}_ X$-modules, see Categories, Section 4.19. Postcomposing with the coprojection $p_ i : \mathcal{F}_ i \to \mathop{\mathrm{colim}}\nolimits _{i' \in I} \mathcal{F}_{i'}$ we get $\alpha = p_ i \circ \alpha _ i$ in the right hand side. We obtain a map

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

Let us show this map is injective. Let $\alpha _ i$ be as above such that $\alpha = p_ i \circ \alpha _ i$ is zero. By the assumption that $\mathcal{G}$ is of finite presentation, for every $x \in X$ we can choose an open neighbourhood $U_ x \subset X$ of $x$ and a finite set $s_{x, 1}, \ldots , s_{x, n_ x} \in \mathcal{G}(U_ x)$ generating $\mathcal{G}|_{U_ x}$. These sections map to zero in the stalk $(\mathop{\mathrm{colim}}\nolimits _{i'} \mathcal{F}_ i)_ x = \mathop{\mathrm{colim}}\nolimits _{i'} \mathcal{F}_{i', x}$. Hence for each $x$ we can pick $i(x) \geq i$ such that after replacing $U_ x$ by a smaller open we have that $s_{x, 1}, \ldots , s_{x, n_ x}$ map to zero in $\mathcal{F}_{i(x)}(U_ x)$. Then $X = \bigcup U_ x$. By condition (3) we can refine this open covering by a finite open covering $X = \bigcup _{j \in J} U_ j$. For $j \in J$ pick $x_ j \in X$ with $U_ j \subset U_{x_ j}$. Set $i' = \max (i(x_ j); j \in J)$. Then $\mathcal{G}|_{U_ j}$ is generated by the sections $s_{x_ j, k}$ which are mapped to zero in $\mathcal{F}_{i(x)}$ and hence in $\mathcal{F}_{i'}$. Hence the composition $\mathcal{G} \to \mathcal{F}_ i \to \mathcal{F}_{i'}$ is zero as desired.

Proof of surjectivity. Let $\alpha$ be an element of the right hand side. For every point $x \in X$ we may choose an open neighbourhood $U \subset X$ and finitely many sections $s_ j \in \mathcal{G}(U)$ which generate $\mathcal{G}$ over $U$ and finitely many relations $\sum f_{kj} s_ j = 0$, $k = 1, \ldots , n$ with $f_{kj} \in \mathcal{O}_ X(U)$ which generate the kernel of $\bigoplus _{j = 1, \ldots , m} \mathcal{O}_ U \to \mathcal{G}$. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ we may assume there exists an index $i \in I$ such that the sections $\alpha (s_ j)$ all come from sections $s_ j' \in \mathcal{F}_ i(U)$. After possibly shrinking $U$ to a smaller open neighbourhood of $x$ and increasing $i$ we may assume the relations $\sum f_{kj} s'_ j = 0$ hold in $\mathcal{F}_ i(U)$. Hence we see that $\alpha |_ U$ lifts to a morphism $\mathcal{G}|_ U \to \mathcal{F}_ i|_ U$ for some index $i \in I$.

By condition (3) and the preceding arguments, we may choose a finite open covering $X = \bigcup _{j = 1, \ldots , m} U_ j$ such that (a) $\mathcal{G}|_{U_ j}$ is generated by finitely many sections $s_{jk} \in \mathcal{G}(U_ j)$, (b) the restriction $\alpha |_{U_ j}$ comes from a morphism $\alpha _ j : \mathcal{G} \to \mathcal{F}_{i_ j}$ for some $i_ j \in I$, and (c) the intersections $U_ j \cap U_{j'}$ are all quasi-compact. For every pair $(j, j') \in \{ 1, \ldots , m\} ^2$ and any $k$ we can find we can find an index $i \geq \max (i_ j, i_{j'})$ such that

$\varphi _{i_ ji}(\alpha _ j(s_{jk}|_{U_ j \cap U_{j'}})) = \varphi _{i_{j'}i}(\alpha _{j'}(s_{jk}|_{U_ j \cap U_{j'}}))$

see Sheaves, Lemma 6.29.1 (2). Since there are finitely many of these pairs $(j, j')$ and finitely many $s_{jk}$ we see that we can find a single $i$ which works for all of them. For this index $i$ all of the maps $\varphi _{i_ ji} \circ \alpha _ j$ agree on the overlaps $U_ j \cap U_{j'}$ as the sections $s_{jk}$ generate $\mathcal{G}$ over this overlap. Hence we get a morphism $\mathcal{G} \to \mathcal{F}_ i$ as desired. $\square$

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.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).