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The Stacks project

Example 6.4.5. Let X be a topological space. For each x \in X suppose given an abelian group M_ x. For U \subset X open we set

\mathcal{F}(U) = \bigoplus \nolimits _{x \in U} M_ x.

We denote a typical element in this abelian group by \sum _{i = 1}^ n m_{x_ i}, where x_ i \in U and m_{x_ i} \in M_{x_ i}. (Of course we may always choose our representation such that x_1, \ldots , x_ n are pairwise distinct.) We define for V \subset U \subset X open a restriction mapping \mathcal{F}(U) \to \mathcal{F}(V) by mapping an element s = \sum _{i = 1}^ n m_{x_ i} to the element s|_ V = \sum _{x_ i \in V} m_{x_ i}. We leave it to the reader to verify that this is a presheaf of abelian groups.


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