The Stacks project

Exercise 111.23.3. Suppose that $A \to B$ is a ring map. Let $f_ i \in A$, $i \in I$ and $g_ j \in B$, $j \in J$ be collections of elements such that

Show that if $A_{f_ i} \to B_{f_ ig_ j}$ is of finite type for all $i, j$ then $A \to B$ is of finite type.