The Stacks project

Exercise 111.37.9. Let $S$ be a graded ring. Let $X = \text{Proj}(S)$. Let $Z, Z' \subset X$ be two closed subschemes. Let $\varphi : Z \to Z'$ be an isomorphism. Assume $Z \cap Z' = \emptyset $. Show that for any $z \in Z$ there exists an affine open $U \subset X$ such that $z \in U$, $\varphi (z) \in U$ and $\varphi (Z \cap U) = Z' \cap U$. (Hint: Use Exercise 111.37.8 and something akin to Schemes, Lemma 26.11.5.)