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

\[ \mathop{\mathrm{Spec}}(A) = \bigcup D(f_ i) \quad \text{and}\quad \mathop{\mathrm{Spec}}(B) = \bigcup D(g_ j). \]

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.

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