Canonical vector space structure on stalk of a presheaf of vector spaces

Question

Let $\mathcal{F}$ be a presheaf of complex vector spaces on $X$. The stalk at $x \in X$ is $$ \mathcal{F}_x = \{ (U, s_U) : s_U \in \mathcal{F}(U) \text{ and } x \in U\}/\sim $$ where $(U,s_U) \sim (V,s_V)$ if $s_U \vert_{U \cap V} = s_V \vert_{U \cap V}$. Supposedly $\mathcal{F}_x$ has a canonical complex vector space structure - I have defined one as setting $[U, s_U] + [V, s_V] = [U, s_U + s_V]$ if $U = V$ and zero otherwise. This is a vector space, but I'm not sure if it's the canonical structure - is it?

Answer 1

You have the right idea but it needs a little modification.

The canonical vector space structure is given by

$$[U, s_U] + r[V, s_V] = [U \cap V, s_U|_{U \cap V} + rs_V|_{U \cap V}]$$

Essentially, you just move to a smaller open set where you can talk about $s_U$ and $s_V$ in the same space. Namely, you are using the vector space structure of $\mathcal{F}(U \cap V)$ to define the thing on the right.