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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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