The Stacks project

Lemma 10.39.6. Let $\{ R_ i, \varphi _{ii'}\} $ be a system of rings over the directed set $I$. Let $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$.

  1. If $M$ is an $R$-module such that $M$ is flat as an $R_ i$-module for all $i$, then $M$ is flat as an $R$-module.

  2. For $i \in I$ let $M_ i$ be a flat $R_ i$-module and for $i' \geq i$ let $f_{ii'} : M_ i \to M_{i'}$ be a $\varphi _{ii'}$-linear map such that $f_{i' i''} \circ f_{i i'} = f_{i i''}$. Then $M = \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i$ is a flat $R$-module.

Proof. Part (1) is a special case of part (2) with $M_ i = M$ for all $i$ and $f_{i i'} = \text{id}_ M$. Proof of (2). Let $\mathfrak a \subset R$ be a finitely generated ideal. By Lemma 10.39.5 it suffices to show that $\mathfrak a \otimes _ R M \to M$ is injective. We can find an $i \in I$ and a finitely generated ideal $\mathfrak a' \subset R_ i$ such that $\mathfrak a = \mathfrak a'R$. Then $\mathfrak a = \mathop{\mathrm{colim}}\nolimits _{i' \geq i} \mathfrak a'R_{i'}$. Since $\otimes $ commutes with colimits the map $\mathfrak a \otimes _ R M \to M$ is the colimit of the maps

\[ \mathfrak a'R_{i'} \otimes _{R_{i'}} M_{i'} \longrightarrow M_{i'} \]

These maps are all injective by assumption. Since colimits over $I$ are exact by Lemma 10.8.8 we win. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 10.39: Flat modules and flat ring maps

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 05UU. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 10.39.6 as pdf