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.)
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