Some basic properties of closed subschemes

Question

I was wondering if someone could clarify me some basic properties of closed subschemes. Let $X$ be a scheme and $Z$ a closed subscheme. Let $W$ be an affine open subset of $X$. Then how does $Z \cap W$ work as a scheme? On one hand $Z \cap W$ is an open subscheme of $Z$. On the other hand $Z \cap W$ is a closed subscheme of $W$. Are these two scheme structures on $Z \cap W$ in fact the same thing?

Answer 1

Suppose $X = \operatorname{Spec} A$ is an affine scheme, $U = D(f)$ is a standard open for some $f \in A$, and $Z = \operatorname{Spec} A/I$ for some ideal $I \subset A$.

Then $U \cap Z$ as an open subset of $Z$ is $\operatorname{Spec} (A/I)_f$. Considered as a closed subset of $U$, one gets $\operatorname{Spec}(A_f / IA_f)$.

But localization and quotient commute, so $(A/I)_f = A_f / IA_f$, and so the two notions coincide.

For general $X$ and $U$, the same should apply by gluing together affine opens.