$\mathcal{O}_X$-modules at the generic fiber

Question

Let $X$ be a scheme and $\eta$ be its generic point. Let $\mathcal{F}$ be an $\mathcal{O}_X$-module such that the stalk $\mathcal{F}_{\eta}$ of $\mathcal{F}$ at the generic point is zero. Then, can we conclude that $\mathcal{F}$ is zero because the closure of $\eta$ is $X$? If not can you give me a counter-example? Thank you!

Answer 1

Let $k$ be a field. $k[x]/(x^2)$ is a $k[x]$-module, so that $\widetilde{(k[x]/(x^2))}$ is a nontrivial $\mathcal O_{\mathbb A_k^1}$-module.

Then, $\widetilde{(k[x]/(x^2))}_{\eta}=(k[x]/(x^2))_{(0)}=0$.