What can we say about a restriction to a closed subset of an affine scheme?

Question

Let $X=\operatorname{Spec}(A)$ and $(X,\mathscr O_X)$ the affine scheme and $Z \subset X$ a closed subset. By definition we have $\alpha \lhd A$ s.t $Z=\operatorname{Spec}(A/\alpha)$ (at least up to the canonical obvious $\pi^*:\operatorname{Spec}(A/\alpha) \rightarrow \operatorname{Spec}(A)$) and let $i:Z\rightarrow X$ be the obvious inclusion. By definition we have $ \mathscr O_{X_{|Z}}=i^{-1}\mathscr O_X=:\mathscr O_Z$ (i.e $\mathscr O_Z(U)= (\lim_{U\subset V}\mathscr O_X(V))^+$ where V is open) (The + denotes the sheafification). Is that true that $(Z,\mathscr O_Z)$ is affine? If so, do we have $Z\simeq \operatorname{Spec}(A/\alpha)$ as affine schemes? And if the answer is yes - How one can be thinking about the that the scheme stracture of $A/\alpha$ can be obtained by the scheme structure of $A$ despite the fact that we define the scheme structure of a commutative ring by its spectrum's open subsets?

Answer 1

The definition of $\mathcal{O}_{Z}$ as $i^{-1}\mathcal{O}_X$ is not correct: $(Z,\mathcal{O}_X|_Z)$ may not even be a scheme! Let $X=\operatorname{Spec} k[x]$ for some field $k$ and let $i:Z\to X$ be the closed immersion of a closed point. Then $(i^{-1}\mathcal{O}_X)(Z) = (i^{-1}\mathcal{O}_X)_{z}\cong \mathcal{O}_{X,z}\cong k[x]_{(x)}$, so if $(Z,\mathcal{O}_X|_Z)$ were to be a scheme, then $Z$ should be isomorphic to $\operatorname{Spec} (\mathcal{O}_X|_Z)(Z)$ because every point in a scheme has an affine neighborhood. But this ring $k[x]_{(x)}$ has two prime ideals $(0)$ and $(x)$, not one prime ideal, as would be required for this to be a scheme.

I don't think there's anything nefarious going on with the follow-up questions you're asking - one should think that on the topological level, $\operatorname{Spec} A/\alpha\to\operatorname{Spec} A$ is a closed immersion, so $\operatorname{Spec} A/\alpha$ has the subspace topology from $A$, and then the way structure sheaves work on affine schemes fills in the rest of the way this has to work. (It's a little hard to tell how much to address these follow-ups, because your underlying premise was wrong, but it looks like you might still have some confusions there even after resolving that first step. ¯\_(ツ)_/¯ )