The Stacks project

Proof. We will use the characterization of Lemma 29.7.5. We have to show that for any open $U \subset X$ the map $\mathcal{O}_ X(U) \to \mathcal{O}_ X(U \cap f^{-1}V)$ is injective. It suffices to prove this when $U$ is an affine open which maps into an affine open $W \subset Y$. Say $W = \mathop{\mathrm{Spec}}(A)$ and $U = \mathop{\mathrm{Spec}}(B)$. Then $V \cap W = D(f_1) \cup \ldots \cup D(f_ n)$ for some $f_ i \in A$, see Algebra, Lemma 10.29.1. Thus we have to show that $B \to B_{f_1} \times \ldots \times B_{f_ n}$ is injective. We are given that $A \to A_{f_1} \times \ldots \times A_{f_ n}$ is injective and that $A \to B$ is flat. Since $B_{f_ i} = A_{f_ i} \otimes _ A B$ we win. $\square$