I'm doing an exercise in algebraic geometry where I have to find an $\mathcal{O}_X$-module $\mathcal{G}$ over an algebraic variety $X$ such that the canonical morphism $$ (\mathcal{G}^{\mathbb{N}})_x\rightarrow (\mathcal{G}_x)^\mathbb{N}$$ Is not injective.
The problem has as hint to consider $X=\mathbb{A}^1(k)$, $x=0$, a sequence $\{a_n\}\subset \mathbb{A}^1\setminus \{0\}$ and the direct sum of the $\mathcal{O}_X$-modules whose support is the point $a_n$. But I can't figure out who are those modules!
If the support of a sheaf of groups $\mathcal{F}$ is defined as the set of points $x\in X$ such that $\mathcal{F}_x\neq 0$. Then I think that a sheaf has a single point as support iff the sheaf is a skyscraper sheaf. But I can't give an structure of $\mathcal{O}_X$-module to an skyscraper sheaf.
I have an example of such a $\mathcal{G}$ when $X$ is a disjoint union of numerable copies of $\mathbb{A}^1$, but this is not an algebraic variety.
If you can give me an example that works or explain me the hint better I will be grateful. Thanks in advance.