The Stacks project

Lemma 6.29.1. Let $X$ be a topological space. Let $I$ be a directed set. Let $(\mathcal{F}_ i, \varphi _{ii'})$ be a system of sheaves of sets over $I$, see Categories, Section 4.21. Let $U \subset X$ be an open subset. Consider the canonical map

\[ \Psi : \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(U) \longrightarrow \left(\mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i\right)(U) \]

  1. If all the transition maps are injective then $\Psi $ is injective for any open $U$.

  2. If $U$ is quasi-compact, then $\Psi $ is injective.

  3. If $U$ is quasi-compact and all the transition maps are injective then $\Psi $ is an isomorphism.

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

Proof. Assume all the transition maps are injective. In this case the presheaf $\mathcal{F}' : V \mapsto \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i(V)$ is separated (see Definition 6.11.2). By the discussion above we have $(\mathcal{F}')^\# = \mathop{\mathrm{colim}}\nolimits _ i \mathcal{F}_ i$. By Lemma 6.17.5 we see that $\mathcal{F}' \to (\mathcal{F}')^\# $ is injective. This proves (1).

Assume $U$ is quasi-compact. Suppose that $s \in \mathcal{F}_ i(U)$ and $s' \in \mathcal{F}_{i'}(U)$ give rise to elements on the left hand side which have the same image under $\Psi $. Since $U$ is quasi-compact this means there exists a finite open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$ and for each $j$ an index $i_ j \in I$, $i_ j \geq i$, $i_ j \geq i'$ such that $\varphi _{ii_ j}(s)$ and $\varphi _{i'i_ j}(s')$ have the same restriction to $U_ j$. Let $i''\in I$ be $\geq $ than all of the $i_ j$. We conclude that $\varphi _{ii''}(s)$ and $\varphi _{i'i''}(s')$ agree on the opens $U_ j$ for all $j$ and hence that $\varphi _{ii''}(s) = \varphi _{i'i''}(s')$. This proves (2).

Assume $U$ is quasi-compact and all transition maps injective. Let $s$ be an element of the target of $\Psi $. Since $U$ is quasi-compact there exists a finite open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$, for each $j$ an index $i_ j \in I$ and $s_ j \in \mathcal{F}_{i_ j}(U_ j)$ such that $s|_{U_ j}$ comes from $s_ j$ for all $j$. Pick $i \in I$ which is $\geq $ than all of the $i_ j$. By (1) the sections $\varphi _{i_ ji}(s_ j)$ agree over the overlaps $U_ j \cap U_{j'}$. Hence they glue to a section $s' \in \mathcal{F}_ i(U)$ which maps to $s$ under $\Psi $. This proves (3).

Assume the hypothesis of (4). In particular we see that $U$ is quasi-compact and hence by (2) we have injectivity of $\Psi $. Let $s$ be an element of the target of $\Psi $. By assumption there exists a finite open covering $U = \bigcup _{j = 1, \ldots , m} U_ j$, with $U_ j \cap U_{j'}$ quasi-compact for all $j, j' \in J$ and for each $j$ an index $i_ j \in I$ and $s_ j \in \mathcal{F}_{i_ j}(U_ j)$ such that $s|_{U_ j}$ is the image of $s_ j$ for all $j$. Since $U_ j \cap U_{j'}$ is quasi-compact we can apply (2) and we see that there exists an $i_{jj'} \in I$, $i_{jj'} \geq i_ j$, $i_{jj'} \geq i_{j'}$ such that $\varphi _{i_ ji_{jj'}}(s_ j)$ and $\varphi _{i_{j'}i_{jj'}}(s_{j'})$ agree over $U_ j \cap U_{j'}$. Choose an index $i \in I$ which is bigger or equal than all the $i_{jj'}$. Then we see that the sections $\varphi _{i_ ji}(s_ j)$ of $\mathcal{F}_ i$ glue to a section of $\mathcal{F}_ i$ over $U$. This section is mapped to the element $s$ as desired. $\square$


Comments (4)

Comment #3641 by Brian Conrad on

In the proof of (4), it should be mentioned that the hypothesis on forces it to be quasi-compact, so injectivity holds by (2).

Comment #8471 by Paul on

I guess in the proof of (2) and (4) the notation was heavily relaxed, since why should hold, actually it holds for some restriction of s and s'. Also it should be or better of some . I recommend to mention it or rework completly the proof.

Comment #9088 by on

OK, I fixed the issue in the proof of (2) but I think the proof of (4) is correct as written now. Change is here.

There are also:

  • 2 comment(s) on Section 6.29: Limits and colimits of sheaves

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 009F. Beware of the difference between the letter 'O' and the digit '0'.