Almost free modules and Mittag--Leffler conditions

Drinfeld recently suggested to replace projective modules by the flat Mittag–Leffler ones in the definition of an infinite dimensional vector bundle on a scheme $X$. Two questions arise: (1) What is the structure of the class $\mathcal{D}$ of all flat Mittag–Leffler modules over a general ring? (2) Can flat Mittag–Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi–coherent sheaves on $X$? We answer (1) by showing that a module $M$ is flat Mittag–Leffler, if and only if $M$ is ${\mathrm{\aleph }}_1$–projective in the sense of Eklof and Mekler. We use this to characterize the rings such that $\mathcal{D}$ is closed under products, and relate the classes of all Mittag–Leffler, strict Mittag–Leffler, and separable modules. Then we prove that the class $\mathcal{D}$ is not deconstructible for any non–right perfect ring. So unlike the classes of all projective and flat modules, the class $\mathcal{D}$ does not admit the homotopy theory tools developed recently by Hovey . This gives a negative answer to (2).