The Stacks project

Exercise 111.51.2. Let $X = \mathop{\mathrm{Spec}}({\mathbf Z}[x, y])$, and let ${\mathcal F}$ be a quasi-coherent ${\mathcal O}_ X$-module. Suppose that ${\mathcal F}$ is zero when restricted to the standard affine open $D(x)$.

1. Show that every global section $s$ of ${\mathcal F}$ is killed by some power of $x$, i.e., $x^ ns = 0$ for some $n\in {\mathbf N}$.

2. Do you think the same is true if we do not assume that ${\mathcal F}$ is quasi-coherent?