Lemma 10.89.11. Let $R \to S$ be a ring map. Let $M$ be an $S$-module. If $S$ is Mittag-Leffler as an $R$-module, and $M$ is flat and Mittag-Leffler as an $S$-module, then $M$ is Mittag-Leffler as an $R$-module.
Proof. We deduce this from the characterization of Proposition 10.89.5. Namely, suppose that $Q_\alpha $ is a family of $R$-modules. Consider the composition
The first arrow is injective as $M$ is flat over $S$ and $S$ is Mittag-Leffler over $R$ and the second arrow is injective as $M$ is Mittag-Leffler over $S$. Hence $M$ is Mittag-Leffler over $R$. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.