Exercise 111.23.1. Suppose that $A$ is a ring and $M$ is an $A$-module. Let $f_ i$, $i \in I$ be a collection of elements of $A$ such that

\[ \mathop{\mathrm{Spec}}(A) = \bigcup D(f_ i). \]

Show that if $M_{f_ i}$ is a finite $A_{f_ i}$-module, then $M$ is a finite $A$-module.

Show that if $M_{f_ i}$ is a flat $A_{f_ i}$-module, then $M$ is a flat $A$-module. (This is kind of silly if you think about it right.)

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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