Intersection of a closed subscheme and an open subscheme of a scheme

Question

Let $X$ be a scheme. Let $Z$ be a closed subscheme of $X$. Let $U$ be an open subscheme of $X$. Then $Y = U \cap Z$ is an open subscheme of $Z$. Can we identify $Y$ with $U\times_X Z$?

Answer 1

Yes. This doesn't have anything to do with closed subscheme. If $p: Z \to X$ is a morphism of schemes and $U \subset X$ is open subscheme, then the fibre product is $p^{-1}(U)$ with open subscheme structure.