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
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
Comment #7965 by Stacks Project on