The Stacks project

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. $\square$

Comments (2)

Comment #7705 by Ryo Suzuki on

This lemma is essenntially a corollary of 6.29.1, because infinite direct sum is filtered colimit of finite direct sums, and transition maps are injecive.

Comment #7965 by on

Although what you say is true, this proof is a lot quicker. So I am going to leave this as is.

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