The Stacks project

Remark 20.24.3. Let $X = \bigcup _{i \in I} U_ i$ be a locally finite open covering. Denote $j_ i : U_ i \to X$ the inclusion map. Suppose that for each $i$ we are given an abelian sheaf $\mathcal{F}_ i$ on $U_ i$. Consider the abelian sheaf $\mathcal{G} = \bigoplus _{i \in I} (j_ i)_*\mathcal{F}_ i$. Then for $V \subset X$ open we actually have

\[ \Gamma (V, \mathcal{G}) = \prod \nolimits _{i \in I} \mathcal{F}_ i(V \cap U_ i). \]

In other words we have

\[ \bigoplus \nolimits _{i \in I} (j_ i)_*\mathcal{F}_ i = \prod \nolimits _{i \in I} (j_ i)_*\mathcal{F}_ i \]

This seems strange until you realize that the direct sum of a collection of sheaves is the sheafification of what you think it should be. See discussion in Modules, Section 17.3. Thus we conclude that in this case the complex of Lemma 20.24.1 has terms

\[ {\mathfrak C}^ p(\mathcal{U}, \mathcal{F}) = \bigoplus \nolimits _{i_0 \ldots i_ p} (j_{i_0 \ldots i_ p})_* \mathcal{F}_{i_0 \ldots i_ p} \]

which is sometimes useful.

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