The Stacks project

Lemma 10.89.9. If $M = \mathop{\mathrm{colim}}\nolimits M_ i$ is the colimit of a directed system of Mittag-Leffler $R$-modules $M_ i$ with universally injective transition maps, then $M$ is Mittag-Leffler.

Proof. Let $(Q_{\alpha })_{\alpha \in A}$ be a family of $R$-modules. We have to show that $M \otimes _ R (\prod Q_\alpha ) \to \prod M \otimes _ R Q_\alpha $ is injective and we know that $M_ i \otimes _ R (\prod Q_\alpha ) \to \prod M_ i \otimes _ R Q_\alpha $ is injective for each $i$, see Proposition 10.89.5. Since $\otimes $ commutes with filtered colimits, it suffices to show that $\prod M_ i \otimes _ R Q_\alpha \to \prod M \otimes _ R Q_\alpha $ is injective. This is clear as each of the maps $M_ i \otimes _ R Q_\alpha \to M \otimes _ R Q_\alpha $ is injective by our assumption that the transition maps are universally injective. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 10.89: Interchanging direct products with tensor

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Your email address will not be published.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 0AS7. Beware of the difference between the letter 'O' and the digit '0'.