# Does Mittag-Lefflerness descend?

I have read in the Stacks Project that if $A \to B$ is a faithfully flat ring homomorphism, $M$ is an $A$-module, and $M \otimes_A B$ is a flat, Mittag-Leffler $B$-module, then $M$ is a flat, Mittag-Leffler $A$-module. It is well known that flatness descends, so I wonder: does Mittag-Lefflerness descend as well (without the flatness condition)?

• It does not answer completely your question but I suggest you to have a look to this paper arxiv.org/pdf/0704.3690v1.pdf In particular Example 1.6 and Proposition 1.7 discuss similar matters – Simone Virili Dec 4 '12 at 19:16