The Stacks project

111.23 Glueing

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

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

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

  2. 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.)

Remark 111.23.2. In algebraic geometric language this means that the property of “being finitely generated” or “being flat” is local for the Zariski topology (in a suitable sense). You can also show this for the property “being of finite presentation”.

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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