  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 $\aleph_1$--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).

Rings and Algebras;Algebraic Geometry