Remark 10.88.8. Let $M$ be a flat $R$-module. By Lazard's theorem (Theorem 10.81.4) we can write $M = \mathop{\mathrm{colim}}\nolimits M_ i$ as the colimit of a directed system $(M_ i, f_{ij})$ where the $M_ i$ are free finite $R$-modules. For $M$ to be Mittag-Leffler, it is enough for the inverse system of duals $(\mathop{\mathrm{Hom}}\nolimits _ R(M_ i, R), \mathop{\mathrm{Hom}}\nolimits _ R(f_{ij}, R))$ to be Mittag-Leffler. This follows from criterion (4) of Proposition 10.88.6 and the fact that for a free finite $R$-module $F$, there is a functorial isomorphism $\mathop{\mathrm{Hom}}\nolimits _ R(F, R) \otimes _ R N \cong \mathop{\mathrm{Hom}}\nolimits _ R(F, N)$ for any $R$-module $N$.

## 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Your email address will not be published.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.

## Comments (0)