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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