Restriction of sheaf on a closed subscheme is in general not quasi-coherent?

Question

In Hartshorne 5.2.4, we have

If $Y$ is a closed subscheme of a scheme $X$, then the sheaf $\mathcal{O}_{X|Y}$ is not in general quasi-coherent in $Y$.

I'm having a hard time believing this, mainly because I can't think of any examples. Would any affine $X$ make an example? Say we have the closed subscheme $Y = \text{Spec } A/\mathfrak{a}$ embedded in $X = \text{Spec } A$.

Thanks!

Answer 1

Let $X=\text{Spec} \mathbb{Z}$, and take $Y$ the closed point $(p)$. Thus $Y=\text{Spec} \mathbb{Z}/p\mathbb{Z}$. Then the sheaf $\mathcal{O}_X|_Y=\mathcal{O}_{X,(p)}=\mathbb{Z}_{(p)}$ since $Y$ is just a closed point. This is cannot be given a $\Gamma(Y,\mathcal{O}_Y)=\mathbb{Z}/p\mathbb{Z}$ module structure. More info can be found: Concrete Example: Subsheaf is not Quasi-Coherent

However, I would be interested in seeing a more geometric example.