Lemma 17.3.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $I$ be a set. For $i \in I$, let $\mathcal{F}_ i$ be a sheaf of $\mathcal{O}_ X$-modules. For $U \subset X$ quasi-compact open the map

is bijective.

Lemma 17.3.5. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $I$ be a set. For $i \in I$, let $\mathcal{F}_ i$ be a sheaf of $\mathcal{O}_ X$-modules. For $U \subset X$ quasi-compact open the map

\[ \bigoplus \nolimits _{i \in I} \mathcal{F}_ i(U) \longrightarrow \left(\bigoplus \nolimits _{i \in I} \mathcal{F}_ i\right)(U) \]

is bijective.

**Proof.**
If $s$ is an element of the right hand side, then there exists an open covering $U = \bigcup _{j \in J} U_ j$ such that $s|_{U_ j}$ is a finite sum $\sum _{i \in I_ j} s_{ji}$ with $s_{ji} \in \mathcal{F}_ i(U_ j)$. Because $U$ is quasi-compact we may assume that the covering is finite, i.e., that $J$ is finite. Then $I' = \bigcup _{j \in J} I_ j$ is a finite subset of $I$. Clearly, $s$ is a section of the subsheaf $\bigoplus _{i \in I'} \mathcal{F}_ i$. The result follows from the fact that for a finite direct sum sheafification is not needed, see Lemma 17.3.2 above.
$\square$

Your email address will not be published.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.

## Comments (2)

Comment #7705 by Ryo Suzuki on

Comment #7965 by Stacks Project on